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Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
These are the initial notes for a class I am preparing for this summer in the Mathematics of Intelligent Systems. we will start with the vectors spaces, their basis and dimensions. The, we will look at one the basic applications the linear regression.
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Vector Space & Sub Space Presentation
Presented By: Sufian Mehmood Soomro
Department: (BS) Computer Science
Course Title: Linear Algebra
Shah Abdul Latif University Ghotki Campus
Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
These are the initial notes for a class I am preparing for this summer in the Mathematics of Intelligent Systems. we will start with the vectors spaces, their basis and dimensions. The, we will look at one the basic applications the linear regression.
I am Manuela B. I am a Linear Algebra Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick. I have been helping students with their assignments for the past 9 years. I solve assignments related to Linear Algebra.
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You can also call on +1 678 648 4277 for any assistance with Linear Algebra Assignment.
Vector Space & Sub Space Presentation
Presented By: Sufian Mehmood Soomro
Department: (BS) Computer Science
Course Title: Linear Algebra
Shah Abdul Latif University Ghotki Campus
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
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 ...
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
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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
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 ...
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
📚 Struggling with Math Assignments? We've got your back! 🧮
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🔹 Algebra, Geometry, Calculus, Statistics, and more!
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🔹 Problem-solving, proofs, equations, and data analysis.
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2️⃣ Fill in the assignment details and requirements.
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4️⃣ Relax while our experts work on your assignment.
5️⃣ Receive your completed assignment, review it, and excel in your academics!
🌐 Don't let math assignments stress you out. Our reliable and affordable math assignment help is just a click away. Trust the experts to guide you toward academic success. Get started now at mathsassignmenthelp.com!
Unlock Your Mathematical Potential with MathAssignmentHelp.com! 🧮✨Maths Assignment Help
Are complex equations and theorems giving you a tough time? Say goodbye to math-related stress because MathAssignmentHelp.com is here to rescue you! 🚀📚
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📊 **2. Homework Assistance:** Don't let assignments weigh you down! We offer top-notch homework help that ensures your work is accurate and submission-ready.
🧭 **3. Exam Prep:** Tackling upcoming exams? Our comprehensive study materials and expert insights will boost your confidence and help you excel.
📈 **4. Concept Clarification:** Whether it's calculus, algebra, statistics, or any other math branch, we specialize in clarifying concepts, filling in knowledge gaps, and helping you truly grasp the subject.
🤝 **Why Choose Us?**
✅ **Qualified Experts:** Our team consists of math aficionados with profound expertise in various mathematical domains.
⏱️ **Timely Assistance:** Tight deadlines? No problem! We're equipped to handle urgent assignments without compromising on quality.
🌐 **User-Friendly Platform:** Our website's intuitive interface ensures a seamless experience from start to finish.
🔒 **100% Confidential:** Your privacy matters. We ensure all your information and interactions with us remain strictly confidential.
🌈 **Embrace the Joy of Learning Math:** Mathematics is not just about numbers; it's about problem-solving, critical thinking, and unlocking new perspectives. Let's make your math journey exciting and fulfilling together!
📣 **Special Offer:** For a limited time, new users get an exclusive discount on their first service. Don't miss out on this opportunity to experience stress-free math learning!
👉 **Visit Us Today:** [MathAssignmentHelp.com](https://www.mathsassignmenthelp.com/)
📱 **Follow Us:** Stay updated with math tips, fun facts, and more on our social media channels! 📚🎉
Let's conquer math together! 🚀🧮
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Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...NelTorrente
In this research, it concludes that while the readiness of teachers in Caloocan City to implement the MATATAG Curriculum is generally positive, targeted efforts in professional development, resource distribution, support networks, and comprehensive preparation can address the existing gaps and ensure successful curriculum implementation.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Calculus Homework Help
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Calculus Homework Help
2. Problem 1: Explain each of the following concepts precisely (yet succinctly):
a) a vector space;
b) a set of linearly independent vectors;
c) the dimension of a vector space;
d) a basis for a vector space;
e) the components of a vector i
f) the scalar or inner product between two vectors;
g) the distance between two vectors;
h) an orthonormal basis;
i) a linear transformation on a vector space;
j) an invariant subspace of a linear transformation;
k) the null space of a linear transformation;
l) a singular linear transformation;
m) the components of a linear transformation in a basis; and
n) the scalar invariants of a linear transformation.
Problem 2: Give an example illustrating each concept in Problem 1.
Problem
Maths Assignment Help
3. Solution
PROBLEM 1:
a) A vector space is a set V of elements called vectors together with operations of addition and
multiplication by a scalar, where these operations must have the following properties:
(A) Corresponding to every pair of vectors x, y ∈ V there is a vector in V, denoted by
x+y, and called the sum of x and y, with the following properties:
(1) x + y = y + x for all x, y ∈ V;
(2) x + (y + z) = (x + y) + z for all x, y, z ∈ V;
(3) there is a unique vector in V, denoted by o and called the null vector, with the
property that x + o = x for all x ∈ V; and
(4) corresponding to every vector x ∈ V there is a unique vector in V, denoted by −x with the
property that x + (−x) = o
(B) Corresponding to every real number α ∈ R and every vector x ∈ V there is a vector in V, denoted by
αx, and called the product of α and x, with the following properties:
(5) α(βx) = (αβ)x for all α, β ∈ R and all x ∈ V;
(6) α(x + y) = αx + αy for all α ∈ R and all x, y ∈ V;
(7) (α + β)x = αx + βx for all α, β ∈ R and all x ∈ V; and
(8) 1x = x for all x ∈ V.
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4. b) A set of vectors {f1, f2, . . . , fn} is said to be linearly independent if the only scalars α1, α2, . . . , αn for
which
α1f1 + α2f2 . . . + αnfn = o
are α1 = α2 = . . . = αn = 0.
c) If a vector space V contains a linearly independent set of n (> 0) vectors but contains no linearly
independent set of n + 1 vectors we say that the dimension of V is n.
d) If V is a n-dimensional vector space then any set of n linearly independent vectors is called a basis
for V.
e) If {f1, f2, . . . , fn} is a basis for an n-dimensional vector space V, then any vector x ∈ V can be
expressed in the form
x = ξ1f1 + ξ2f2 + . . . + ξnfn
where the set of scalars ξ1, ξ2, . . . , ξn is unique and are called the components of x in the basis {f1, f2, . .
. , fn}.
f) To every pair of vectors x, y ∈ V we associate a real number denoted by x · y and called the scalar
product of x and y provided that this product has the following properties:
(9) x · y = y · x for all x, y ∈ V;
(10) (x + y) · z = x · z + y · z for all x, y, z ∈ V;
(11) (αx) · y = α(x · y) for all α ∈ R and all vectors x, y ∈ V;
and
(12) x · x > 0 for all vectors x = o in V.
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5. g) The real number denoted by |x − y| and defined as |x − y| = (x − y) · (x − y) is called the distance
between the vectors x and y.
h) If {e1, e2, . . . , en} is a basis for an n-dimensional vector space and if
we say that {e1, e2, . . . , en} is an orthonormal basis.
i) A linear transformation A on a vector space V is a transformation that assigns to each vector x ∈ V
a unique vector in V which we denote by Ax with the properties:
(13) A(x + y) = Ax + Ay for all vectors x, y ∈ V; and
(14) A(αx) = α(Ax) for every α ∈ R and every vector x ∈ V.
j) Let S be a subset of a vector space V. Suppose further that S itself is in fact a vector space on its own
right under the same operations of addition and scalar multiplication as in V. Then S is said to be a
subspace of V. Finally, suppose in addition that Ax ∈ S for all x ∈ S. Then we say that S is an invariant
subspace of A.
k) The set N of all vectors x for which Ax = o is called the null space of A.
l) A linear transformation A is said to be singular if there is a vector x = o for which Ax = o.
m) The n2 real numbers Aij defined by
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6. are called the components of the linear transformation A in the basis {e1, e2, . . . , en}.
n) A scalar valued function φ defined on the set of all linear transformations is said to be a scalar invariant if
φ(QAQT ) = φ(A) for every linear transformation A and all orthogonal linear transformations Q.
PROBLEM 2:
a) Consider the set V of all 2 × 2 matrices x of the form
where x1 and x2 range over all real numbers; let
be the null vector; and define addition, x + y, and scalar multiplication, αx, in the natural way by
One can verify that all of the requirements (1)–(8) of Problem 1 are satisfied by these operations, and moreover,
that x + y and αx are both in V when x, y ∈ V and α ∈ R. Thus V is a vector space.
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7. b) Consider the following two vectors f1 and f2:
One can readily verify that if α1f1+α2f2 = o, then necessarily α1+2α2 = 0 and 2α1+α2 = 0 which in turn
implies that α1 = α2 = 0. Thus {f1, f2} is a linearly independent set of vectors.
c) Consider the following three vectors f1, f2, x,
where x is an arbitrary vector in V. One can readily verify that if α1f1 + α2f2 + α3x = o then necessarily α1 +
2α2 + x1α3 = 0 and 2α1 + α2 + x2α3 = 0. Observe that the choice
satisfies these two scalar equations. Thus if α1f1 + α2f2 + α3x = o this does not require that all the α’s
vanish and so {f1, f2, x} is a linearly dependent set of vectors. Recall that {f1, f2} is a linearly independent
set of vectors. Thus the dimension of V is 2.
d) Since V is a 2-dimensional vector space and since the set of vectors {f1, f2} is linearly independent, it
follows that {f1, f2} is a basis for V.
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8. e) Consider the basis {f1, f2} and let x be an arbitrary vector in V. Then one can readily verify that
are the components of x in the basis {f1, f2}
f) Corresponding to any two vectors x, y ∈ V, where
tentatively define their scalar product as
x · y = x1y1 + x2y2.
One can verify that this definition satisfies all of the requirement (9)–(12) of Problem 1 and therefore is in
fact a legitimate definition of a scalar product.
g) The distance between the two vectors
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9. h) Consider the two vectors
Observe that e1 · e2 = 0, |e1| = |e2| = 1 and so {e1, e2} forms an orthonormal basis for V.
i) Consider a transformation A that takes the vector
One can verify that the requirements (13), (14) of Problem 1 are satisfied, and moreover that Ax ∈ V for
all x ∈ V. Therefore A is a linear transformation.
j) Consider the set S of all vectors x of the form
where x ranges over all real numbers. Clearly S is a subset of V. Moreover, one can verify that S itself is a
vector space on its own right under the same operations of addition and scalar multiplication as in V. Thus
S is a subspace of V. Furthermore, observe that Ax = x for all vectors x ∈ S, so that in particular Ax ∈ S for
all x ∈ S. Thus S is an invariant subspace of A. (In fact it is a one-dimensional invariant subspace
associated with the eigenvalue +1).
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10. k) From item (i) we see that if Ax = o then necessarily x = o. Thus the null space of A is comprised of a single
vector, the null vector: N = {o}.
l) As noted in the preceding item, Ax = o implies that necessarily x = o. Therefore A is nonsingular.
m) Observe from the definitions of A, e1 and e2 that Ae1 = e2 and Ae2 = e1. Thus the components of A in the
basis {e1, e2} are
n) Consider the scalar-valued function φ(A) = det A defined for all linear transformations A. Then for any linear
transformation A and any orthogonal transformation Q we have φ(QAQT ) = det(QAQT ) = det(Q) det(A) det(QT ) =
det(Q) det(A) det(Q) = (±1)2 det A = det A. Thus the function φ(A) = det A has the property that φ(QAQT ) = φ(A) for
every linear transformation A and all orthogonal linear transformations Q. Thus det A is a scalar invariant of A.
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