This document provides notes on vector spaces, which are fundamental objects in linear algebra. It begins with examples of vector spaces such as R2, R3, C2, C3 and defines vector spaces more generally as sets that are closed under vector addition and scalar multiplication and satisfy other properties like the existence of additive identities. It then provides several examples of vector spaces including the set of all n-tuples over a field, the set of all m×n matrices, the set of differentiable functions on an interval, and the set of polynomials with coefficients in a field.
The document defines a subspace as a non-empty subset W of a vector space V that is itself a vector space under the operations defined on V. It notes that every vector space has at least two subspaces: itself and the zero subspace containing only the zero vector. To prove that W is a subspace of V, we only need to verify that W is closed under the vector space operations. Examples are provided to illustrate this, such as showing that the set W={(x,0,0)| x in R} is a subspace of R3 by verifying it is closed under vector addition and scalar multiplication.
Chapter 4: Vector Spaces - Part 1/Slides By PearsonChaimae Baroudi
This document defines vectors and vector spaces. It begins by defining vectors in 2D and 3D space as matrices and describes operations like addition, scalar multiplication, and subtraction. It then defines a vector space as a set of vectors that satisfies 10 axioms related to these operations. Examples of vector spaces include the set of 2D and 3D vectors, sets of matrices, and sets of polynomials. The document also defines subspaces and proves that the span of a set of vectors in a vector space forms a subspace.
This document provides information about vector spaces and subspaces. It defines a vector space as a set of objects called vectors that can be added together and multiplied by scalars, subject to certain rules. A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The null space of a matrix is the set of solutions to the homogeneous equation Ax=0 and is a subspace. The column space of a matrix is the set of all linear combinations of its columns and is also a subspace. Examples are provided to illustrate these concepts.
(1) The document discusses inner product spaces and related linear algebra concepts such as orthogonal vectors and bases, Gram-Schmidt process, orthogonal complements, and orthogonal projections.
(2) Key topics covered include defining inner products and their properties, finding orthogonal vectors and constructing orthogonal bases, using Gram-Schmidt process to orthogonalize a set of vectors, defining and finding orthogonal complements of subspaces, and computing orthogonal projections of vectors.
(3) Examples are provided to demonstrate computing orthogonal bases, orthogonal complements, and orthogonal projections in inner product spaces.
The document defines key concepts in vector spaces including vector space, subspace, span of a set of vectors, and basis. It provides examples to illustrate these concepts. Specifically:
- A vector space is a set of objects called vectors that can be added together and multiplied by scalars, satisfying certain properties.
- A subspace is a subset of a vector space that is itself a vector space under the operations of the original space.
- The span of a set of vectors S is the set of all possible linear combinations of the vectors in S.
- A basis is a set of vectors that spans a vector space and is linearly independent. It provides a standard representation for vectors in the space.
This document discusses vector spaces and subspaces. It begins by defining a vector space as a set V with two operations, vector addition and scalar multiplication, that satisfy certain properties. Examples of vector spaces include R2 and the space of real polynomials of degree n or less.
It then defines a subspace as a subset of a vector space that is itself a vector space under the inherited operations. For a subset to be a subspace, it must be closed under vector addition and scalar multiplication, and contain the zero vector. Examples given include lines and planes through the origin in R3.
The span of a set S of vectors is defined as the set of all linear combinations of the vectors in S, and it
The document defines a subspace as a non-empty subset W of a vector space V that is itself a vector space under the operations defined on V. It notes that every vector space has at least two subspaces: itself and the zero subspace containing only the zero vector. To prove that W is a subspace of V, we only need to verify that W is closed under the vector space operations. Examples are provided to illustrate this, such as showing that the set W={(x,0,0)| x in R} is a subspace of R3 by verifying it is closed under vector addition and scalar multiplication.
Chapter 4: Vector Spaces - Part 1/Slides By PearsonChaimae Baroudi
This document defines vectors and vector spaces. It begins by defining vectors in 2D and 3D space as matrices and describes operations like addition, scalar multiplication, and subtraction. It then defines a vector space as a set of vectors that satisfies 10 axioms related to these operations. Examples of vector spaces include the set of 2D and 3D vectors, sets of matrices, and sets of polynomials. The document also defines subspaces and proves that the span of a set of vectors in a vector space forms a subspace.
This document provides information about vector spaces and subspaces. It defines a vector space as a set of objects called vectors that can be added together and multiplied by scalars, subject to certain rules. A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The null space of a matrix is the set of solutions to the homogeneous equation Ax=0 and is a subspace. The column space of a matrix is the set of all linear combinations of its columns and is also a subspace. Examples are provided to illustrate these concepts.
(1) The document discusses inner product spaces and related linear algebra concepts such as orthogonal vectors and bases, Gram-Schmidt process, orthogonal complements, and orthogonal projections.
(2) Key topics covered include defining inner products and their properties, finding orthogonal vectors and constructing orthogonal bases, using Gram-Schmidt process to orthogonalize a set of vectors, defining and finding orthogonal complements of subspaces, and computing orthogonal projections of vectors.
(3) Examples are provided to demonstrate computing orthogonal bases, orthogonal complements, and orthogonal projections in inner product spaces.
The document defines key concepts in vector spaces including vector space, subspace, span of a set of vectors, and basis. It provides examples to illustrate these concepts. Specifically:
- A vector space is a set of objects called vectors that can be added together and multiplied by scalars, satisfying certain properties.
- A subspace is a subset of a vector space that is itself a vector space under the operations of the original space.
- The span of a set of vectors S is the set of all possible linear combinations of the vectors in S.
- A basis is a set of vectors that spans a vector space and is linearly independent. It provides a standard representation for vectors in the space.
This document discusses vector spaces and subspaces. It begins by defining a vector space as a set V with two operations, vector addition and scalar multiplication, that satisfy certain properties. Examples of vector spaces include R2 and the space of real polynomials of degree n or less.
It then defines a subspace as a subset of a vector space that is itself a vector space under the inherited operations. For a subset to be a subspace, it must be closed under vector addition and scalar multiplication, and contain the zero vector. Examples given include lines and planes through the origin in R3.
The span of a set S of vectors is defined as the set of all linear combinations of the vectors in S, and it
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
The document discusses vector spaces and related concepts:
1) It defines a vector space as a set V with vector addition and scalar multiplication operations that satisfy certain properties. Examples of vector spaces include R2, the plane in R3, and the space of real polynomials.
2) A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication and thus forms a vector space with the inherited operations. Examples given include the x-axis in Rn and solution spaces of linear differential equations.
3) The span of a set of vectors is the smallest subspace that contains those vectors, consisting of all possible linear combinations of the vectors in the set.
Liner algebra-vector space-1 introduction to vector space and subspace Manikanta satyala
This document discusses the key differences between scalar and vector quantities. Scalars only have magnitude, while vectors have both magnitude and direction. It then defines vector spaces as sets of vectors that are closed under vector addition and scalar multiplication. Examples of vector spaces include n-dimensional spaces, matrix spaces, polynomial spaces, and function spaces. Subspaces are also introduced as vector spaces that are subsets of a larger vector space and satisfy the same properties.
This document discusses normed vector spaces and related concepts. It introduces the definition of a norm on a vector space and properties like the triangle inequality. It then extends topological concepts like open and closed sets to normed vector spaces. Examples of normed vector spaces include function spaces like C[a,b] equipped with the supremum norm. The document also discusses concepts like convergence in normed spaces and dense subsets, with examples involving polynomial approximation of continuous functions.
The document provides an overview of vector spaces and related linear algebra concepts. It defines vector spaces, subspaces, basis, dimension, and rank. Key points include:
- A vector space is a set that is closed under vector addition and scalar multiplication. It must satisfy certain axioms.
- A subspace is a subset of a vector space that is also a vector space.
- A basis is a minimal set of linearly independent vectors that span the entire vector space. The dimension of a vector space is the number of vectors in its basis.
- The rank of a matrix is the number of linearly independent rows in its row-reduced echelon form. It provides a measure of the matrix's linear
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
1) An inner product space is a vector space with an inner product defined that satisfies certain properties like linearity and positive-definiteness.
2) The Gram-Schmidt process is used to transform a basis into an orthogonal basis and then an orthonormal basis by successively subtracting projections.
3) The angle between two vectors in an inner product space can be computed using the inner product and the norms of the vectors.
The document discusses vector spaces and related linear algebra concepts. It defines vector spaces and lists the axioms that must be satisfied. Examples of vector spaces include the set of all pairs of real numbers and the space of 2x2 symmetric matrices. The document also discusses subspaces, linear combinations, span, basis, dimension, row space, column space, null space, rank, nullity, and change of basis. It provides examples and explanations of these fundamental linear algebra topics.
1) The document discusses directional derivatives and the gradient of functions of several variables. It defines the directional derivative Duf(c) as the slope of the function f in the direction of the unit vector u at the point c.
2) It shows that the partial derivatives of f can be computed by treating all but one variable as a constant. The gradient of f is defined as the vector of its partial derivatives.
3) It derives an expression for the directional derivative Duf(c) in terms of the partial derivatives of f and the components of the unit vector u, showing the relationship between directional derivatives and the gradient.
This document discusses the gamma and beta functions. It defines the gamma function and lists some of its key properties. Examples are provided to demonstrate how to evaluate integrals using gamma function properties. The beta function is then defined and its relationship to the gamma function explained. Dirichlet's integral theorem and its extension to multiple dimensions is covered. Applications to finding volumes and masses are demonstrated. References for further reading on gamma and beta functions are listed at the end.
linear transformation and rank nullity theorem Manthan Chavda
In these notes, I will present everything we know so far about linear transformations.
This material comes from sections in the book, and supplemental that
I talk about in class.
1. The document introduces vectors and matrices as ways to collectively represent multiple quantities or relationships between quantities.
2. Vectors are used to represent positions, food orders, prices, and other grouped data. Matrices are used to represent ingredient amounts for different foods and connections between rooms in a floorplan.
3. All of the examples can be expressed using vectors and matrices, with the key information being the numbers in the vectors and matrices.
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
1. Quiz 4 will cover sections 3.3, 5.1, and 5.2 and will be on Thursday, February 18.
2. To find the nth power of a matrix A that has been diagonalized as A = PDP-1, one raises the diagonal elements of D to the nth power to obtain Dn, leaving P and P-1 unchanged.
3. A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, allowing it to be written as A = PDP-1, where the columns of P are the eigenvectors and the diagonal elements of D are the corresponding eigenvalues.
The document discusses two methods for evaluating integrals: integration by substitution and integration by parts. Integration by substitution involves setting up an integral in a way that allows substituting a new variable u for an expression involving x, making the integral easier to evaluate. Integration by parts is a method for evaluating integrals of products of functions by breaking it into multiple integrals using the formula ∫u v dx = u∫v dx −∫u' (∫v dx) dx. The document provides examples of applying both methods to evaluate integrals of trigonometric, logarithmic, and exponential functions. It also briefly mentions partial fractions as a method to decompose rational functions into simpler fractions.
This document provides an overview of vector spaces and related concepts such as linear combinations, spans, bases, and subspaces. Some key points:
- A vector space is a set equipped with vector addition and scalar multiplication satisfying certain properties. Examples include Rm and the space of polynomials.
- A linear combination of vectors is a sum of the form v = x1v1 + x2v2 + ... + xnvn. The span of vectors is the set of all their linear combinations.
- A set of vectors is linearly independent if the only way to get the zero vector as a linear combination is with all scalars equal to zero.
- A basis is a linearly independent set of vectors
Euler's Method is used to approximate solutions to differential equations. The document provides two examples:
1) Approximating y(2) given dy/dx = 2x + y, y(1) = -3, using two steps of size 0.5. The approximation is y(2) ≈ -3.75.
2) Approximating y(4) given dy/dx = y - 2, y(0)=4, using four steps of size 1. The approximation is y(4) ≈ 34.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
The document discusses various methods to compute the rank of a matrix:
1) Using Gauss elimination, where the rank is the number of pivot columns in the echelon form of the matrix.
2) Using determinants of sub-matrices (minors), where the rank is the largest order of a non-zero minor.
3) Transforming the matrix to normal form using row and column operations, where the rank is the number of non-zero rows of the resulting identity matrix.
Worked examples are provided to illustrate computing the rank of matrices using these different methods.
This document provides an introduction to functions and limits. It defines key concepts such as domain, range, and different types of functions including algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate how to find the domain and range of functions, evaluate functions, and draw graphs of functions. Function notation and the concept of a function as a rule that assigns each input to a single output are also explained.
This document provides an introduction to functions and their key concepts. It defines what a function is, using examples to illustrate functions that relate variables. Functions have a domain and range, and can be represented graphically. Common types of functions are discussed, including algebraic functions like polynomials and rational functions, as well as trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Methods for determining a function's domain and range and drawing its graph are presented.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
The document discusses vector spaces and related concepts:
1) It defines a vector space as a set V with vector addition and scalar multiplication operations that satisfy certain properties. Examples of vector spaces include R2, the plane in R3, and the space of real polynomials.
2) A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication and thus forms a vector space with the inherited operations. Examples given include the x-axis in Rn and solution spaces of linear differential equations.
3) The span of a set of vectors is the smallest subspace that contains those vectors, consisting of all possible linear combinations of the vectors in the set.
Liner algebra-vector space-1 introduction to vector space and subspace Manikanta satyala
This document discusses the key differences between scalar and vector quantities. Scalars only have magnitude, while vectors have both magnitude and direction. It then defines vector spaces as sets of vectors that are closed under vector addition and scalar multiplication. Examples of vector spaces include n-dimensional spaces, matrix spaces, polynomial spaces, and function spaces. Subspaces are also introduced as vector spaces that are subsets of a larger vector space and satisfy the same properties.
This document discusses normed vector spaces and related concepts. It introduces the definition of a norm on a vector space and properties like the triangle inequality. It then extends topological concepts like open and closed sets to normed vector spaces. Examples of normed vector spaces include function spaces like C[a,b] equipped with the supremum norm. The document also discusses concepts like convergence in normed spaces and dense subsets, with examples involving polynomial approximation of continuous functions.
The document provides an overview of vector spaces and related linear algebra concepts. It defines vector spaces, subspaces, basis, dimension, and rank. Key points include:
- A vector space is a set that is closed under vector addition and scalar multiplication. It must satisfy certain axioms.
- A subspace is a subset of a vector space that is also a vector space.
- A basis is a minimal set of linearly independent vectors that span the entire vector space. The dimension of a vector space is the number of vectors in its basis.
- The rank of a matrix is the number of linearly independent rows in its row-reduced echelon form. It provides a measure of the matrix's linear
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
1) An inner product space is a vector space with an inner product defined that satisfies certain properties like linearity and positive-definiteness.
2) The Gram-Schmidt process is used to transform a basis into an orthogonal basis and then an orthonormal basis by successively subtracting projections.
3) The angle between two vectors in an inner product space can be computed using the inner product and the norms of the vectors.
The document discusses vector spaces and related linear algebra concepts. It defines vector spaces and lists the axioms that must be satisfied. Examples of vector spaces include the set of all pairs of real numbers and the space of 2x2 symmetric matrices. The document also discusses subspaces, linear combinations, span, basis, dimension, row space, column space, null space, rank, nullity, and change of basis. It provides examples and explanations of these fundamental linear algebra topics.
1) The document discusses directional derivatives and the gradient of functions of several variables. It defines the directional derivative Duf(c) as the slope of the function f in the direction of the unit vector u at the point c.
2) It shows that the partial derivatives of f can be computed by treating all but one variable as a constant. The gradient of f is defined as the vector of its partial derivatives.
3) It derives an expression for the directional derivative Duf(c) in terms of the partial derivatives of f and the components of the unit vector u, showing the relationship between directional derivatives and the gradient.
This document discusses the gamma and beta functions. It defines the gamma function and lists some of its key properties. Examples are provided to demonstrate how to evaluate integrals using gamma function properties. The beta function is then defined and its relationship to the gamma function explained. Dirichlet's integral theorem and its extension to multiple dimensions is covered. Applications to finding volumes and masses are demonstrated. References for further reading on gamma and beta functions are listed at the end.
linear transformation and rank nullity theorem Manthan Chavda
In these notes, I will present everything we know so far about linear transformations.
This material comes from sections in the book, and supplemental that
I talk about in class.
1. The document introduces vectors and matrices as ways to collectively represent multiple quantities or relationships between quantities.
2. Vectors are used to represent positions, food orders, prices, and other grouped data. Matrices are used to represent ingredient amounts for different foods and connections between rooms in a floorplan.
3. All of the examples can be expressed using vectors and matrices, with the key information being the numbers in the vectors and matrices.
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
1. Quiz 4 will cover sections 3.3, 5.1, and 5.2 and will be on Thursday, February 18.
2. To find the nth power of a matrix A that has been diagonalized as A = PDP-1, one raises the diagonal elements of D to the nth power to obtain Dn, leaving P and P-1 unchanged.
3. A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, allowing it to be written as A = PDP-1, where the columns of P are the eigenvectors and the diagonal elements of D are the corresponding eigenvalues.
The document discusses two methods for evaluating integrals: integration by substitution and integration by parts. Integration by substitution involves setting up an integral in a way that allows substituting a new variable u for an expression involving x, making the integral easier to evaluate. Integration by parts is a method for evaluating integrals of products of functions by breaking it into multiple integrals using the formula ∫u v dx = u∫v dx −∫u' (∫v dx) dx. The document provides examples of applying both methods to evaluate integrals of trigonometric, logarithmic, and exponential functions. It also briefly mentions partial fractions as a method to decompose rational functions into simpler fractions.
This document provides an overview of vector spaces and related concepts such as linear combinations, spans, bases, and subspaces. Some key points:
- A vector space is a set equipped with vector addition and scalar multiplication satisfying certain properties. Examples include Rm and the space of polynomials.
- A linear combination of vectors is a sum of the form v = x1v1 + x2v2 + ... + xnvn. The span of vectors is the set of all their linear combinations.
- A set of vectors is linearly independent if the only way to get the zero vector as a linear combination is with all scalars equal to zero.
- A basis is a linearly independent set of vectors
Euler's Method is used to approximate solutions to differential equations. The document provides two examples:
1) Approximating y(2) given dy/dx = 2x + y, y(1) = -3, using two steps of size 0.5. The approximation is y(2) ≈ -3.75.
2) Approximating y(4) given dy/dx = y - 2, y(0)=4, using four steps of size 1. The approximation is y(4) ≈ 34.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
The document discusses various methods to compute the rank of a matrix:
1) Using Gauss elimination, where the rank is the number of pivot columns in the echelon form of the matrix.
2) Using determinants of sub-matrices (minors), where the rank is the largest order of a non-zero minor.
3) Transforming the matrix to normal form using row and column operations, where the rank is the number of non-zero rows of the resulting identity matrix.
Worked examples are provided to illustrate computing the rank of matrices using these different methods.
This document provides an introduction to functions and limits. It defines key concepts such as domain, range, and different types of functions including algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate how to find the domain and range of functions, evaluate functions, and draw graphs of functions. Function notation and the concept of a function as a rule that assigns each input to a single output are also explained.
This document provides an introduction to functions and their key concepts. It defines what a function is, using examples to illustrate functions that relate variables. Functions have a domain and range, and can be represented graphically. Common types of functions are discussed, including algebraic functions like polynomials and rational functions, as well as trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Methods for determining a function's domain and range and drawing its graph are presented.
This document provides an introduction to functions and their key concepts. It defines a function as a rule that assigns each element in one set to a unique element in another set. Functions can be represented graphically and algebraically. Common types of functions discussed include polynomial, linear, constant, rational, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate domain, range, and graphing of different function types.
I am Fabian H. I am a Calculus Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics, Deakin University, Australia. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
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Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
The document discusses linear algebra concepts of vector spaces and bases. It introduces vector spaces as sets of objects that can be added and multiplied by field elements. Subspaces are defined as vector space subsets that are also vector spaces. Linear combinations are expressed as combinations of basis vectors with field coefficients. A basis is defined as a linearly independent set of vectors that span the vector space. Dimensions refer to the number of basis vectors needed to represent elements of the vector space.
Math 511 Problem Set 4, due September 21Note Problems 1 tAbramMartino96
Math 511 Problem Set 4, due September 21
Note: Problems 1 through 7 are the ones to be turned in. The remainder of the problems are
for extra functional analytic goodness.
1. Fix a,b ∈ R with a < b. Show that {1, t, t2, . . . , tn} is a linearly independent subset of
C[a,b]. From this conclude that {1, t, t2, t3, . . .} is a linearly independent set in C[a,b]. Give
an example of a function f ∈ C[a,b] so that f /∈ span{1, t, t2, . . .}.
2. Prove that if 1 ≤ p1 ≤ p2 ≤∞ then lp1 ⊆ lp2 .
3. Consider C[0, 2] with the function ‖ ·‖1 defined by
‖f‖1 =
∫ 2
0
|f(x)|dx, for f ∈ C[0, 2].
(a) Prove that ‖ ·‖1 is a norm.
(b) Prove that the normed linear space (C[0, 2],‖·‖1) is not complete (and thus not a Banach
space) by considering the sequence of functions
fn(x) =
1, x ≤ 1 − 1
n
n−nx, 1 − 1
n
< x < 1 + 1
n
−1, x ≥ 1 + 1
n
.
Show these are continuous functions, this sequence is a Cauchy sequence in the metric
derived from ‖ ·‖1, but that this sequence does not converge in C[0, 2] with this metric.
4. Let V be a vector space over R or C. A subset A ⊆ V is convex if for any v,w ∈ A and any
λ ∈ [0, 1] then λv + (1 −λ)w ∈ A, i.e. the segement connecting v and w is also in A.
(a) Let W be a vector subspace of V . Show that W is convex.
(b) Let X be a normed linear space. Show that the unit ball B1(0) is convex.
5. show that c ⊆ l∞ is a vector subspace of l∞ (see 1.5-3 for the definition of c) and so is c0, the
set of all sequences (xn) so that limn→∞ xn = 0.
6. Let 1 ≤ p < ∞ and en ∈ lp be the sequence with 1 in the nth place and 0 in all othe coordinates.
Show that {en : n ∈ N} is a Schauder basis for lp.
7. Now if X is a Banach space and (yn) a sequence in X, prove that
∑∞
n=1 ‖yn‖ < ∞ does imply
the convergence of
∑∞
n=1 yn. Thus in Banach spaces, absolute convergence implies convergence
of the series.
The following questions are for you to think about and not to be turned in.
1001. What is the completion of (0, 1) as a metric subspace of R with the euclidean metric?
Explain.
1002. Show that the discrete metric on a nontrivial vector space cannot be obtained from a norm.
1003. Show that if a normed vector space has a Schauder basis, then the space is separable. (You
can use a similar argument to your proof that lp is separable for 1 ≤ p < ∞.)
1004. Prove the general Hölder inequality: Suppose 1 ≤ r < p < ∞, and assume that
1
p
+
1
q
=
1
r
.
Show that for x = (x1,x2, . . .) and y = (y1,y2, . . .), and if we define the componentwise product
xy = (x1y1,x2y2, . . .), then
‖xy‖r ≤‖x‖p‖y‖q.
You may assume that x ∈ lp and y ∈ lq, although this is not necessary. (Hint: 1 = 1p
r
+ 1q
r
, and
use the regular Hölder inequality on particular sequences).
(Note: We can extend this to let p = r, and in this case q = ∞. The result will still hold.)
1005. Give an example of a subspace of l∞ which is not closed. Repeat for l2. (Hint: Look at
problem 3, p. 70)
1006. Let X be a normed vector space. Show that the convergenc ...
This document defines functions and how to prove they are one-to-one (1-1) and onto. It provides definitions of 1-1, onto, and bijective functions. It then gives examples of proving a function is onto and 1-1. To prove onto, one shows that every element in the target set is mapped to by solving the function rule for the domain variable. To prove 1-1, one assumes two domain elements map to the same target and shows they must be equal.
This document introduces complex numbers by representing real numbers as vectors on a number line. It defines the imaginary unit i as the operator that rotates a vector 90 degrees, making it possible to take the square root of negative numbers. Complex numbers are then represented as points in a plane, allowing them to be written as z = x + iy, where x is the real component and y is the imaginary component. Key properties of complex numbers like addition and multiplication are explained geometrically and through polar coordinates, with the unit circle playing an important role.
This document provides lecture notes for a complex functions course at the University of Bristol. It includes:
1) A reading list recommending textbooks on complex analysis.
2) Information about course structure including homework assignments, problem classes, and math cafe sessions.
3) An introduction to complex numbers covering definitions of addition, multiplication, and properties like conjugates and moduli. Geometric interpretations of complex numbers as points in the complex plane are also discussed.
4) Explanations of key concepts in complex analysis like roots of complex numbers, the polar form of complex numbers, and geometric interpretations of addition and multiplication. Regions in the complex plane corresponding to subsets of complex numbers are briefly mentioned.
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The document defines a vector space and its properties. A vector space is a set V in which vectors can be added and multiplied by scalars, while satisfying certain axioms. Some key points:
- Rn is the vector space of all n-dimensional real vectors. Examples include R2 for the 2D plane and R3 for 3D space.
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This document provides an introduction to set theory and functions of a single variable in mathematics. It defines key concepts like sets, subsets, integers, rational numbers, irrational numbers, and real numbers. It explains how sets like the integers, rationals, and irrationals combine to form the set of real numbers. It also defines what a function is and provides examples of functions from real numbers to real numbers. Functions are described as mappings between sets that assign a unique output to each input. Graphing functions is discussed as representing the set of points formed by the inputs and outputs of a function.
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
The document discusses using Plücker coordinates to determine if a set of homogeneous polynomials generate the vector space of higher degree polynomials when projected onto a quotient ring. It begins by defining Plücker coordinates and showing that for two quadratic generators, the cubic vector space has zero image if the Plücker quantity P1P3 - P22 is non-zero. It then aims to generalize this to three cubic generators and the quartic vector space.
This document contains exercises related to dynamical systems and periodic points. It includes the following summaries:
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2. The map f(x)=|x-2| has a fixed point at x=1. Other periodic and pre-periodic points are [0,2]\{1\} of period 2 and (-∞,0)∪(2,+∞) which are pre-periodic.
3. Expanding maps of the circle are topologically mixing since intervals get longer under iteration, eventually covering the entire circle.
The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
The document defines and provides properties of various mathematical functions including:
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- Functions including domain, co-domain, range, and the number of possible functions between sets.
- Types of functions such as polynomial, algebraic, transcendental, rational, exponential, logarithmic, and absolute value functions.
- Graphs of important functions are shown such as 1/x, sinx, logx, |x|, [x], and their key properties are described.
2. Linear Algebra for Machine Learning: Basis and DimensionCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the second part which is discussing basis and dimension.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Quasar Chunawala is seeking a position as a quantitative modeller. He has a Bachelor's degree in Mathematics and Information Technology. His self-learning projects include implementing a cubic spline interpolation algorithm in Python and summarizing the Vanna-Volga method for pricing FX implied volatility smiles. Currently, he works as a Quantitative Analyst at Credit Suisse, where he performs quantitative analysis tasks related to rates and credit derivatives.
This document describes pricing options using lattice models, specifically binomial trees. It provides details on:
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The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise boosts blood flow and levels of neurotransmitters and endorphins which elevate and stabilize mood.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
This document describes an algorithm to compute the nth Fibonacci number using recursive squaring in better than linear time. It does this by raising the matrix ((1,1),(1,0)) to the nth power using recursive squaring. If n is even, it recursively computes An/2 and multiplies it by itself. If n is odd, it recursively computes A(n-1)/2 and multiplies it by itself and the original matrix A. This allows it to compute An in O(log n) time rather than the naive O(n) linear time approach. Pseudocode and a C++ implementation are provided.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise stimulates the production of endorphins in the brain which elevate mood and reduce stress levels.
On building FX Volatility surface - The Vanna Volga methodQuasar Chunawala
The document discusses methods for constructing FX volatility surfaces, specifically the Vanna-Volga method. It begins by explaining the volatility smile seen in FX option markets and reasons for its existence. It then provides details on implementing the bisection method to find implied volatility. Next, it shows an example of plotting the volatility skew of NIFTY options. Finally, it provides the key equations for the Vanna-Volga method, including defining the strikes for ATM, 25 delta call and put, and deriving the expressions for vega, vanna, and volga of calls and puts.
Interpolation techniques - Background and implementationQuasar Chunawala
This document discusses interpolation techniques, specifically Lagrange interpolation. It begins by introducing the problem of interpolation - given values of an unknown function f(x) at discrete points, finding a simple function that approximates f(x).
It then discusses using Taylor series polynomials for interpolation when the function value and its derivatives are known at a point. The error in interpolation approximations is also examined.
The main part discusses Lagrange interpolation - given data points (xi, f(xi)), there exists a unique interpolating polynomial Pn(x) of degree N that passes through all the points. This is proved using the non-zero Vandermonde determinant. Lagrange's interpolating polynomial is then introduced as a solution.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
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Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
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Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
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The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
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With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
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When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
The binding of cosmological structures by massless topological defects
Vector spaces
1. Vector Spaces• January 2018 • Linear Algebra notes
Vector Spaces
Quasar Chunawala, Mumbai
January 2018
Abstract
These are my notes on vector spaces, one of the most fundamental objects in linear algebra and the
whole of mathematics.
Contents
1 Vector spaces 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Conclusion 8
1. Vector spaces
1.1. Preliminaries
Let’s refresh what R2, R3, C2, C3 mean.
The vector space R2, which you can think of as a plane, consists of all ordered pairs of real
numbers.
R2
= {(x, y) : x, y ∈ R}
The vector space R3, which you can think of as ordinary space, consists of all ordered triples
of real numbers.
R3
= {(x, y, z) : x, y, z ∈ R}
The vector space Rn, which you can think of as n-dimensional space, consists of all possible
ordered lists of length n of real numbers. Such an ordered collection of n elements is called n-tuple.
Rn is the set of all such n-tuples.
Rn
= {(x1, x2, . . . , xn) : x1, x2, . . . , xn ∈ R}
If n ≥ 4, we cannot easily visualize Rn as a physical object. The same problem arises if we
work with complex numbers. C1 can be thought of as a plane. But, consider C2 defined as,
C2
= {(z1, z2) : z1, z2 ∈ C}
For n ≥ 2, the human brain cannot provide geometric models of Cn. However, even if n is
large, we can treat them as geometric vectors and perform algebraic manipulations in Fn, as easily
as in R2 or R3, as we will study shortly.
1
2. Vector Spaces• January 2018 • Linear Algebra notes
Often the mathematics of Fn becomes cleaner, if we use a single entity to denote an n-tuple,
without explicitly writing its co-ordinates. For example, suppose the rule of addition is defined on
Fn by adding elements coordinate wise.
(x1, x2, . . . , xn) + (y1, y2, . . . , yn) = (x1 + y1, x2 + y2, . . . , xn + yn)
It is convenient if we represent
x = (x1, x2, x3, . . . , xn)
and call it a vector.
Thus, commutative property of addition in Fn should be expressed as
x + y = y + x
We begin with the basic concept of linear algebra. For the definition that follows, we assume
that we are given a particular field F, the scalars to be used are elements of F.
1.2. Definition.
A vector space (or a linear space) is a set V of elements satisfying the following axioms -
1. There is a function called vector addition, that assign to every pair of elements x, y ∈ V, an
element x + y in V called the sum of x and y, such that :
(a) Addition is commutative.
x + y = y + x
(b) Addition is associative.
x + (y + z) = (x + y) + z
(c) Existence of a zero vector.
There exists in V, a unique vector 0, called the 0 vector such that:
x + 0 = x, ∀x ∈ V
(d) Existence of a negative element.
To every vector x in V, there corresponds a unique vector −x, called the negative of x,
such that
x + (−x) = 0
where 0 is the 0 vector.
2. There is a function called scalar multiplication, that assigns to every pair α ∈ F and x ∈ V,
α · x in V, called the scalar product of α and x, in such a way that :
(a) Multiplication by scalars is associative.
α(βx) = (αβ)x
(b) Existence of a unit scalar.
1x = x
3. Distributive properties.
2
3. Vector Spaces• January 2018 • Linear Algebra notes
(a) Scalar multiplication distributes over addition.
α(x + y) = αx + αy
(b) Scalar addition distributes over multiplication.
(α + β)x = αx + βx
The relation between the vector space V and the underlying field F is usually described by
saying that V is a vector space over F. If F is the field of real numbers, V is called a real vector
space, similarly if F is Q or C, we speak of rational vector spaces or complex vector spaces.
This axiomatic definition opens up a whole new world of things we can call vectors now.
For example, n- tuples, m × n matrices, complex numbers, the solutions of a linear differential
equation, polynomials are all vectors. Vectors are not necessarily elements of R2 or geometric
vectors.
1.3.
1. Let V = R = F. Define addition and multiplication in the usual way, real numbers are
added and multiplied by scalars. As elements in V are real numbers, and the set of real
numbers forms a field, the elements V satisfy all axioms of a vector space. Thus R is a vector
space over R.
2. Let V = C = F. Addition and multiplication are to be ordinary complex number addition
and multiplication. Show that, C is a vector space over C.
Proof. We know that, if z1, z2 ∈ C, =⇒ z1 + z2 ∈ C. Therefore, V is closed under addition.
• As C is a field, addition of complex numbers is commutative.
z1 + z2 = z2 + z1
• As C is a field, addition of complex numbers is associative.
z1 + (z2 + z3) = (z1 + z2) + z3
• Existence of zero vector. C is field and contains the zero vector 0 := 0 + i0, such that
z + 0 = z
• Existence of negative element. If z = x + iy ∈ C, there exists a unique element
−z = −x − iy ∈ C, such that,
z + (−z) = (x + iy) + (−x − iy) = (x − x) + i(y − y) = 0
We also know that if α ∈ C, x ∈ C, αx ∈ C. Hence, V is closed with respect to scalar
multiplication.
• Existence of unit element. As C is a field, it contains a multiplicative identity 1. Thus,
1z = z
holds.
3
4. Vector Spaces• January 2018 • Linear Algebra notes
• Multiplication is associative. This holds as multiplication in C is associative.
α(βz) = (αβ)z
We also verify the distributive properties.
• Scalar multiplication distributes well over vector addition.
α(z1 + z2) = (α1 + iα2)(x1 + x2 + i(y1 + y2)
= α1(x1 + x2) − α2(y1 + y2) + iα1(y1 + y2) + iα2(x1 + x2)
= (α1x1 − α2y1) + i(α2x1 + α1y1) + (α1x2 − α2y2) + i(α2x2 + α1y2)
= (α1 + iα2)(x1 + iy1) + (α1 + iα2)(x2 + iy2)
= αz1 + αz2
• On similar lines, addition distributes well over multiplication.
(α + β)z = αz + βz
Thus, C is a vector space over C.
In general, any field is a vector space over itself. F(F) is a vector space.
3. Let V = F2 be the set of all column vectors which have just two components(co-ordinates).
F2
= {
x1
x2
: x1, x2 ∈ F}
Let x, y ∈ F2. Then, x =
x1
x2
and y =
y1
y2
. And let α ∈ F.
Define addition function as:
x + y =
x1 + x2
y1 + y2
Define scalar multiplication as:
αx =
αx1
αx2
It is an easy exercise to prove that F2 is a vector space over F, with respect to addition and
scalar multiplication defined above.
4. All n- tuples of real numbers form the vector space Rn over the real numbers R.
5. All n-tuples of complex numbers form the vector space Cn over the complex numbers C.
It is an easy exercise to prove that Fn is a vector space over F with respect to addition and
scalar multiplication defined in the usual way.
6. Let V = Fm×n be the set of all m × n matrices.
Fm×n
=
A =
a11 a12 . . . a1n
a21 a22 . . . a2n
...
...
am1 am2 . . . amn
: aij ∈ F
4
5. Vector Spaces• January 2018 • Linear Algebra notes
Let x, y be vectors in Fm×n. We define vector addition as ordinary matrix addition element-
wise.
x + y =
x11 + y11 x12 + y12 . . . x1n + y1n
x21 + y21 x22 + y22 . . . x2n + y2n
...
...
xm1 + ym1 xm2 + ym2 . . . xmn + ymn
We define scalar multiplication as
αx =
αx11 αx12 . . . αx1n
αx21 αx22 . . . αx2n
...
...
αxm1 αxm2 . . . αxmn
All m × n matrices form the vector space Fm×n over the field F, with respect to addition and
scalar multiplcation defined above.
Convenient notation:
C[a, b] - The set of all real valued functions defined from [a, b] onto F.
Ck(a, b) - The set of all k-times continuously differentiable functions from (a, b) onto F.
7. Let C([a, b]) be the set of all real valued functions defined from [a, b] onto R.
Let p, q be two functions in C([a, b]). Define addition as a new function p + q that assigns to
each t ∈ [a, b], the value p(t) + q(t).
(p + q)(t) = p(t) + q(t)
Define multiplication as a new function αp that assigns to each t ∈ [a, b], the value αp(t).
It is an easy exercise to prove that C([a, b]) is a vector space over R.
8. Let Ck((a, b)) be the set of all real valued functions f with the property that
dk f
dtk is continuous
in (a, b). Define vector addition and scalar multiplication as before. It is an easy exercise to
prove that Ck((a, b)) is a vector space over R.
We can do a little bit more general. One looks at C∞((a, b)).
9. Let C∞((a, b)) be the set of all real valued functions that are infinitely many times dif-
ferentiable in the open interval (a, b). The same operations of vector addition and scalar
multiplication will tell you that this is a real vector space.
10. Collect all functions f defined from (a, b) onto R in the set
F((a, b)) = {f : (a, b) → R}
In this we look at the set
V = {f ∈ F :
b
a
f (t)dt exists}
5
6. Vector Spaces• January 2018 • Linear Algebra notes
that is the set of all functions that are Riemann integrable. Then, with respect to usual
addition and scalar multiplication, we can show that V is a real vector space.
If f is Riemann-integrable and g is Riemann-integrable, then (f + g) is Riemann integrable.
If f is Riemann-integrable and α is a real number, then α f is Riemann-integrable.
Polynomial functions:
A function p : F → F is a polynomial with coefficients a0, a1, a2, . . . , an ∈ F of degree n, if to
each x ∈ F, p assigns the value p(x) where
p(x) = a0 + a1x + a2x2
+ . . . + anxn
11. Let Pn(R) be set of all polynomials in real variable x with real coefficients of degree not
exceeding n.
Let p, q be two polynomials in Pn(R). Define sum of two polynomials as a new polynomial
(p + q)(x).
(p + q)(x) = p(x) + q(x)
= p0x + p1x + p2x2
+ . . . + pnxn
+ q0x + q1x + q2x2
+ . . . + qnxn
= (p0 + q0) + (p1 + q1)x + (p2 + q2)x2
+ . . . + (pn + qn)xn
We can easily see, that P(R) is closed under polynomial addition defined in the above way.
Clearly, polynomial addition is associative, commutative. There exists a 0 polynomial, such
that 0 + p = 0. There is unique negative polynomial −p = −p0 − p1x − p2x2 − . . . − pnxn,
such that p + (−p) = 0.
If c ∈ R, define scalar multiplication as a new polynomial cp.
(cp)(x) = cp(x)
= cp0 + cp1x + . . . + cpnxn
If we set c = 1, 1 · p = p. Scalar multiplication is associative. Further, the distributive
properties are also satisfied.
With these definitions of addition and scalar multiplication, Pn(R) is vector space over R.
12. Let A be a m × n matrix with its entries as real numbers. That is, A ∈ Rm×n. Consider the
system of m linear equations
Ax = 0
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7. Vector Spaces• January 2018 • Linear Algebra notes
Remember A is a m × n matrix, x =
x1
x2
...
xn
. Collect the set of all solution vectors in a set V
which is a subset of Rn. Rn already has vector addition and scalar multiplication defined. It
can be shown that V is a vector space over R with respect to those operations.
V = {x ∈ Rn
: Ax = 0} ⊆ R
Proof. Let us first show that V is closed with respect to addition. If x, y ∈ V that x, y satisfy
Ax = 0 and Ay = 0, then we must prove that x + y ∈ V that is, x + y satisfied A(x + y) = 0.
We know that matrix multiplication is distributive.
A(x + y) = Ax + Ay
= 0 + 0 = 0
Thus, (x + y) satisfies A(x + y) = 0.
Similarly, if x ∈ V that is x satisfies Ax = 0, then αx ∈ V, that is αx satisfies A(αx) = 0. This
is true, as
A(αx) = α(Ax)
= α(0) = 0
13. Let us look at an ordinary differential equation of the form
F(x, y, y , . . . , y(n)
) = a0(x)
dny
dxn
+ a1(x)
dn−1y
dxn−1
+ an(x) = 0
A function f (x) is the solution of this differential equation if and only if,
F[x, f (x), f (x), . . . , f (n)
(x)] = 0
Collect all such functions f (x) in a set V.
V = {f : F[x, f (x), f (x), . . . , f (n)
(x)] = 0}
It is an easy exercise to prove that the solutions of a linear homogenous differential equation
form a vector space V over the field R.
14. Let V ⊆ R2 be defined as
V := {(x1, x2) : x2 = 5x1 and x1, x2 ∈ R}
These are the set of all points on the straight line y = 5x in a plane. We define vector addition
and scalar multiplication in V to be the usual operations in R2.
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8. Vector Spaces• January 2018 • Linear Algebra notes
x + y = (x1 + y1, x2 + y2)
αx = (αx1, αx2)
Show that V is a vector space.
Proof.
2. Conclusion
“I always thought something was fundamentally wrong with the universe” [? ]
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