The document provides an overview of linear algebra and matrices. It discusses scalars, vectors, matrices, and various matrix operations including addition, subtraction, scalar multiplication, and matrix multiplication. It also covers topics such as identity matrices, inverse matrices, determinants, and using matrices to solve systems of simultaneous linear equations. Key concepts are illustrated with examples throughout.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
formulation of first order linear and nonlinear 2nd order differential equationMahaswari Jogia
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
These Slides are very usefull interms of engineering and as well as in other fields of Study .. These are Related with linear Algebra and there Properties Methods to find out the unknowns from the equation...
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
formulation of first order linear and nonlinear 2nd order differential equationMahaswari Jogia
• Equations which are composed of an unknown function and its derivatives are called differential equations.
• Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
• When a function involves one dependent variable, the equation is called an ordinary differential equation (ODE).
• A partial differential equation (PDE) involves two or more independent variables.
Figure 1: CHARACTERIZATION OF DIFFERENTIAL EQUATION
FIRST ORDER DIFFERENTIAL EQUATION:
FIRST ORDER LINEAR AND NON LINEAR EQUATION:
A first order equation includes a first derivative as its highest derivative.
- Linear 1st order ODE:
Where P and Q are functions of x.
TYPES OF LINEAR DIFFERENTIAL EQUATION:
1. Separable Variable
2. Homogeneous Equation
3. Exact Equation
4. Linear Equation
i. SEPARABLE VARIABLE:
The first-order differential equation:
Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y.
Suppose we can write the above equation as
We then say we have “separated” the variables. By taking h(y) to the LHS, the equation becomes:
Integrating, we get the solution as:
Where c is an arbitrary constant.
EXAMPLE 1.
Consider the DE :
Separating the variables, we get
Integrating we get the solution as:
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
These Slides are very usefull interms of engineering and as well as in other fields of Study .. These are Related with linear Algebra and there Properties Methods to find out the unknowns from the equation...
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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LinearAlgebra.ppt
1. Linear Algebra and Matrices
Methods for Dummies
21st October, 2009
Elvina Chu & Flavia Mancini
2. Talk Outline
• Scalars, vectors and matrices
• Vector and matrix calculations
• Identity, inverse matrices & determinants
• Solving simultaneous equations
• Relevance to SPM
Linear Algebra & Matrices, MfD 2009
3. Scalar
• Variable described by a single number
e.g. Intensity of each voxel in an MRI scan
Linear Algebra & Matrices, MfD 2009
4. Vector
• Not a physics vector (magnitude, direction)
• Column of numbers e.g. intensity of same voxel at
different time points
Linear Algebra & Matrices, MfD 2009
3
2
1
x
x
x
5. Matrices
• Rectangular display of vectors in rows and columns
• Can inform about the same vector intensity at different
times or different voxels at the same time
• Vector is just a n x 1 matrix
4
7
6
1
4
5
3
2
1
A
Square (3 x 3) Rectangular (3 x 2) d i j : ith row, jth column
Defined as rows x columns (R x C)
8
3
7
2
4
1
C
33
32
31
23
22
21
13
12
11
d
d
d
d
d
d
d
d
d
D
Linear Algebra & Matrices, MfD 2009
6. Matrices in Matlab
• X=matrix
• ;=end of a row
• :=all row or column
9
8
7
6
5
4
3
2
1
Subscripting – each element of a matrix
can be addressed with a pair of numbers;
row first, column second (Roman Catholic)
e.g. X(2,3) = 6
X(3, :) =
X( [2 3], 2) =
9
8
7
8
5
“Special” matrix commands:
• zeros(3,1) =
• ones(2) =
• magic(3) =
1
1
1
1
0
0
0
Linear Algebra & Matrices, MfD 2009
2
9
4
7
5
3
6
1
8
11. Matrix Multiplication
“When A is a mxn matrix & B is a kxl matrix,
AB is only possible if n=k. The result will be
an mxl matrix”
A1 A2 A3
A4 A5 A6
A7 A8 A9
A10 A11 A12
m
n
x
B13 B14
B15 B16
B17 B18
l
k
Number of columns in A = Number of rows in B
= m x l matrix
Linear Algebra & Matrices, MfD 2009
12. Matrix multiplication
• Multiplication method:
Sum over product of respective rows and columns
1 0
2 3
1
3
1
2
22
21
12
11
c
c
c
c
1)
(3
1)
(2
3)
(3
2)
(2
1)
(0
1)
(1
3)
(0
2)
(1
5
13
1
2
X =
=
=
Define output
matrix
A B
• Matlab does all this for you!
• Simply type: C = A * B
Linear Algebra & Matrices, MfD 2009
13. Matrix multiplication
• Matrix multiplication is NOT commutative
• AB≠BA
• Matrix multiplication IS associative
• A(BC)=(AB)C
• Matrix multiplication IS distributive
• A(B+C)=AB+AC
• (A+B)C=AC+BC
Linear Algebra & Matrices, MfD 2009
14. Vector Products
Inner product XTY is a scalar
(1xn) (nx1)
Outer product XYT is a matrix
(nx1) (1xn)
xyT
x1
x2
x3
y1 y2 y3
x1y1 x1y2 x1y3
x2y1 x2y2 x2y3
x3y1 x3y2 x3y3
i
i
i
T
y
x
y
x
y
x
y
x
y
y
y
x
x
x
3
1
3
3
2
2
1
1
3
2
1
3
2
1
y
x
Inner product = scalar
Two vectors:
3
2
1
x
x
x
x
3
2
1
y
y
y
y
Outer product = matrix
Linear Algebra & Matrices, MfD 2009
15. Identity matrix
Is there a matrix which plays a similar role as the number 1 in number
multiplication?
Consider the nxn matrix:
For any nxn matrix A, we have A In = In A = A
For any nxm matrix A, we have In A = A, and A Im = A (so 2 possible matrices)
Linear Algebra & Matrices, MfD 2009
16. Identity matrix
1 2 3 1 0 0 1+0+0 0+2+0 0+0+3
4 5 6 X 0 1 0 = 4+0+0 0+5+0 0+0+6
7 8 9 0 0 1 7+0+0 0+8+0 0+0+9
Worked example
A I3 = A
for a 3x3 matrix:
• In Matlab: eye(r, c) produces an r x c identity matrix
Linear Algebra & Matrices, MfD 2009
17. Matrix inverse
• Definition. A matrix A is called nonsingular or invertible if there
exists a matrix B such that:
• Notation. A common notation for the
inverse of a matrix A is A-1. So:
• The inverse matrix is unique when it exists. So if A is invertible, then A-1
is also invertible and then (AT)-1 = (A-1)T
1 1 X
2
3
-1
3
=
2 + 1
3 3
-1 + 1
3 3
= 1 0
-1 2
1
3
1
3
-2+ 2
3 3
1 + 2
3 3
0 1
• In Matlab: A-1 = inv(A) •Matrix division: A/B= A*B-1
Linear Algebra & Matrices, MfD 2009
18. Matrix inverse
• For a XxX square matrix:
• The inverse matrix is:
Linear Algebra & Matrices, MfD 2009
• E.g.: 2x2 matrix
19. Determinants
• Determinants are mathematical objects that are very useful in the
analysis and solution of systems of linear equations (i.e. GLMs).
• The determinant is a function that associates a scalar det(A) to every
square matrix A.
– Input is nxn matrix
– Output is a single
number (real or
complex) called the
determinant
Linear Algebra & Matrices, MfD 2009
20. Determinants
•Determinants can only be found for square matrices.
•For a 2x2 matrix A, det(A) = ad-bc. Lets have at closer look at that:
• A matrix A has an inverse matrix A-1 if and only if det(A)≠0.
• In Matlab: det(A) = det(A)
Linear Algebra & Matrices, MfD 2009
a b
c d
det(A) = = ad - bc
[ ]
21. Solving simultaneous equations
For one linear equation ax=b where the unknown is x and a
and b are constants,
3 possibilities:
If 0
a then b
a
a
b
x 1
thus there is single solution
If 0
a , 0
b then the equation b
ax becomes 0
0
and any value of x will do
If 0
a , 0
b then b
ax becomes b
0 which is a
contradiction
Linear Algebra & Matrices, MfD 2009
22. With >1 equation and >1 unknown
• Can use solution from the single
equation to solve
• For example
• In matrix form
b
a
x 1
1
2
5
3
2
2
1
2
1
x
x
x
x
A X = B
1
5
2
1
3
2
2
1
x
x
X =A-1B
Linear Algebra & Matrices, MfD 2009
4
1
23. • X =A-1B
• To find A-1
• Need to find determinant of matrix A
• From earlier
(2 -2) – (3 1) = -4 – 3 = -7
• So determinant is -7
bc
ad
d
c
b
a
A
)
det(
2
1
3
2
Linear Algebra & Matrices, MfD 2009
A1
1
det(A)
d b
c a
25. Linear Algebra & Matrices, MfD 2009
How are matrices relevant
to fMRI data?
26. Normalisation
Statistical Parametric Map
Image time-series
Parameter estimates
General Linear Model
Realignment Smoothing
Design matrix
Anatomical
reference
Spatial filter
Statistical
Inference
RFT
p <0.05
Linear Algebra & Matrices, MfD 2009
27. Linear Algebra & Matrices, MfD 2009
BOLD signal
Time
single voxel
time series
Voxel-wise time series analysis
Model
specification
Parameter
estimation
Hypothesis
Statistic
SPM
28. How are matrices relevant to fMRI data?
=
a
m
b3
b4
b5
b6
b7
b8
b9
+
e
= b +
Y X ´
Linear Algebra & Matrices, MfD 2009
GLM equation
N
of
scans
29. How are matrices relevant to fMRI data?
Y
Response variable
e.g BOLD signal at a particular
voxel
A single voxel sampled at successive
time points.
Each voxel is considered as
independent observation.
Preprocessing
...
Intens
ity
Ti
me
Y
Time
Y = X . β + ε
Linear Algebra & Matrices, MfD 2009
30. How are matrices relevant to fMRI data?
a
m
b3
b4
b5
b6
b7
b8
b9
b
X ´
Explanatory variables
– These are assumed to be
measured without error.
– May be continuous;
– May be dummy,
indicating levels of an
experimental factor.
Y = X . β + ε
Solve equation for β – tells us
how much of the BOLD signal is
explained by X
Linear Algebra & Matrices, MfD 2009
31. In Practice
• Estimate MAGNITUDE of signal changes
• MR INTENSITY levels for each voxel at
various time points
• Relationship between experiment and
voxel changes are established
• Calculation and notation require linear
algebra
Linear Algebra & Matrices, MfD 2009
32. Summary
• SPM builds up data as a matrix.
• Manipulation of matrices enables unknown
values to be calculated.
Y = X . β + ε
Observed = Predictors * Parameters + Error
BOLD = Design Matrix * Betas + Error
Linear Algebra & Matrices, MfD 2009
33. References
• SPM course http://www.fil.ion.ucl.ac.uk/spm/course/
• Web Guides
http://mathworld.wolfram.com/LinearAlgebra.html
http://www.maths.surrey.ac.uk/explore/emmaspages/option1.ht
ml
http://www.inf.ed.ac.uk/teaching/courses/fmcs1/
(Formal Modelling in Cognitive Science course)
• http://www.wikipedia.org
• Previous MfD slides
Linear Algebra & Matrices, MfD 2009