MATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE
MATRICES AND MATRIX OPERATIONS
SYSTEM OF LINEAR EQUATIONS DETERMINANTS
VECTORS IN 2-SPACE AND 3-SPACE
GENERAL VECTOR SPACESINNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR TRANSFORMATIONS
DIFFERENTIAL EQUATIONS SYSTEM
WEEK CONTENTS (MID EXAM)
1 - Matrix and Matrix ’s Operations
- Elementary Row Operations 2 - System of linear equations
- Gauss-Jordan Elimination 3 - Homogenous system
- Inverse matrix 4 - Determinant - Cofactor expansion, Row Reduction Methods 5 - Crammer Method - Least square Method 6 - Dot Product, orthogonal projection
- Cross product 7 - Space and subspace
WEEK CONTENTS (FINAL EXAM)
8 - Linear independence
- Linear Combination - Basis and Dimension 9 - Basis of Subspace - Basis of Column space, Row space 10 - Inner Product Space : Norm, angle and distance
- Orthogonal and orthonormal set, projection
- Gramm-Schimdt method 11 - Linear Transformation - Transformation matrix 12 - Kernel and Range of T 13 - Eigen Value, Eigen Vector - Diagonalization 14 - System of Differential equations
MATRICES z
Matrix Notation Definition A matrix is a rectangular array of numbers. The numbers in the array are called entries in the matrix. The entry in row i and column j is denoted by the symbol a ij
Size of matrix is described as in terms of the number of rows and
columns it contains A General m x n matrix is written as
mn m m n n a a a a a a a a a
K M M M M K K
2
1
2
22
21
1
12
11
MATRICES z
Matrix Notation
mn m m n n a a a a a a a a a
2
1
2
22
21
1
12
11 Row 1
Column 2 Entry row m and column j
MATRICES
=
2
5
2
1
4
3
3
3
=
22 ,…,a nn
11 ,a
Square Matrix of order n a matrix with n rows and n columns main diagonal : a
1 A z
3
2
4
1 B Order 2 Order 3 Main diagonal
MATRICES [ ]
I z
2 2 x
1
1
=
Zero Matrices A matrix all of whose entries are zero
Identity Matrices A Square matrix with 1’s on the main diagonal and 0’s off the main diagonal. Identity matrix is denoted by I z
3 x
3
1
1
1
=
I
MATRICES z
11 a a a a a a a a a a
12
13
14
22
23
24
33
31
44
21
Triangular Matrices A Square matrix in which all the entries above the main diagonal are zero (lower triangular)
22
31
32
33
41
42
43
44
A Square matrix in which all the entries below the main diagonal are zero (upper triangular )
11 a a a a a a a a a a lower triangular 4x4 upper triangular 4x4
MATRICES z
Reduced row-echelon form Matrices Properties of Reduced row-echelon form
1. If a row does not consist entirely of zeros, then the first non-zero number in the row is 1. We call this (number 1) a leading 1
2. If there are any row that consist entirely of zeros (not-null row), then
they grouped together at the bottom of the matrix3. In any two successive not-null row, the leading 1 in the lower row occurs farther to the right than the leading 1in the higher row
4. Each column that contain leading 1 has zeros everywhere else We will solve a system of linear equations (next chapter) easier when the augmented matrices in reduced row echelon form. Matrix has only properties 1,2 and 3 is called has row-echelon form
MATRICES z
1
2
1
1
2
Reduced row-echelon form Matrices Example of reduced row-echelon matrices
2
1
3
1
2
1
1
1
2
1
3
2
1 Example of matrices not in reduced row-echelon form
1
1
2
3
1 Properties 1
1
2
1
1
1
2
1
3
2
2
Properties 3 Properties 4
Operations on Matrices Addition and Subtraction matrix Scalar Multiples Multiplying Matrices Transpose of a Matrix Trace of a Matrix Elementary Row Operations
2 Example : subtraction
6
6
4
5
3
4
2
3
1
− − − −
=
4
4
4
2
5
3
3
1
− − − −
=
2
2
2
−
8
1
3
Operations on Matrices Addition and Subtraction Definition If A and B are matrices of the same size, then the sum A+B is the matrix obtained by adding entries of B corresponding entries of A
matrix and the difference A −B is the matrix obtained by subtracting
entries of B corresponding entries of A matrix .Example : addition
1
3
3
5
5
3
-
6
6
2
4
4
4
4
6
2
=
10
=
Operations on Matrices Definition If A is any matrix and k is any scalar, then the product kA is the matrix obtained by multiplying each entry of the matrix A by k
3
6
9
12
15
18
=
3
1
2
4
Example : scalar multiples
5
6
3
3 1 .
3 2 .
3 3 .
3 4 .
3 5 .
= 6 .
3 Scalar Multiples
- a
- …+ a
6
4
1
3
2
2
1
5
1
4
3
2
1
=
47
20
20
2 1 .
8
Operations on Matrices Definition
If A is m x r matrix and B is r x n matrix, then the product A x B is the
m x n whose entries are determined as follows. The entry in row i and
column j of AB given by formula (AB) ij= a i1 b
1j
i2 b
2j
ir b rj
Example : multiplying matrices Multiplying Matrices
6 3 .
3 2 .
5 2 .
4 1 .
6 2 .
5 1 .
4 4 .
3 3 .
2 2 .
1 1 .
- = 4 .
Operations on Matrices Definition If A is any m x n matrix, then the transpose of A, denoted by A
1 T A
4
2
5
3
6
=
Transpose of a Matrix
T , is defined to be the n x m matrix that the results from interchanging the rows and columns of A Example : transpose of a matrix
2
3
1 A
4
5
6
=
Operations on Matrices Definition If A is n x n square matrix, then the trace of A, denoted by tr(A), is defined to be the sum of the entries on the main diagonal of A, or given by formula tr(A) = a
+…+a
11 +a
22
nn Example : trace of a matrix
Trace of a Matrix tr(A) = 1+5+9=15
4
7
1 A
9
3
8
5
2
=
6
Operations on Matrices Elementary Row Operations (ERO) Elementary row operations are operations to eliminate matrix to be reduced row-echelon form. When we have the (augmented matrix) reduced row-echelon form, we will get solutions of system of linear equations easier. We will discuss this more at the next chapter There are three types of operations
1. Multiply a row through by a nonzero constant
2. Interchange two rows
3. Add a multiple of one row to another row
Operations on Matrices Steps in elimination (Create reduced row-echelon form) We have matrix A mxn
Go to first row, Change entry a to be 1 (choose the simplest •
11 operation)
- Change entries a , a ,..,a to be 0
21 31 m1 Go to first next row, Change entry a to be 1 (we pass this step •
22 when a =0 and go to the next entry a )
22 2k
Change entries a , a ,..,a • to be 0
1k 3k mk
- Repeat step 3 and 4 until the last row (we will get reduced row-
echelon form)
3 A
11 not leading 1 not zero * we also can choose another operation
1
8
2 1 r r ↔
Operations on Matrices Example elimination using ERO
3
1
1
2
1 3 r r
3
1
Reduced row echelon form ?
3 A
1
1
8
1
1
3
2
1
1
7
=
1
- −
~
Add -3 x row 1 to the second row * Leading 1 a
1 Interchange row 1 and row 2 *
=
2
1
1
2
1
1
7
− −
7
1
8
1
7
1
1
2
1
1
1
3
3
2
1
~
3
3
Operations on Matrices Example elimination using ERO (2) ~
1 Add -2 x row 1 to the third row
3
1 2 r r
1
1
1
3
3 3 r r
2
1
1
- −
- −
2
1
1
1
4
2
1
1 2 r r
7
2
7
2
1
7
2 r r
3
1
1
− − −
1
1
~
1
1
− −
7
1
2
1
1
2
1
− − −
Add -3 x row 3 to the second row
2
- −
1 Leading 1
-
1
Add 1 x row 2 to the third row Multiple row 3 by -1
~ 3 r − Add -1 x row 2 to the first row
~
1
2
4
1
3
− −
2
− − −
1
1
2
4
1
3
− − − −
1
4
~
1
7
Operations on Matrices Example elimination using ERO (3) Add 2 x row 3 to the second row Add -2 x row 3 to the first row
2 r r
3
1
1
2
1
1
7
2
3
2 r r
2
- −
1 Leading 1
-
1
~
1. All ERO notations above is given to help students in understanding this material. We don’t have to write this notations at the next chapter
Reduced row echelon form Notes
3
1
~
2
1
1
2
2
4
1
3
− −
3
1
2. We can also group some ERO notations to make shortly
Operation in Matrices Properties of Matrix Operations
a. A+B=B+A
b. A+(B+C)=(A+B)+C
c. A(BC)=(AB)C
d. A(B+C)=AB+AC
e. k(AB)=(kA)B ; k : skalar T T
f. (A ) =A T T T
g. (AB) =B A
Exercises Consider these matrices
1
2
1
2
1
2
2
2
1
1
3 − −
A
2
3
1 C
E
3 D B
= =
2 1 = = =
3
1
2
4
−
−
2
3
4
1 1
1
2
1. Compute the following
a. BC
b. A – BC
c. CE
Td. CB
e. D – CB
f. EE – A
2. Which matrices below are in reduced row echelon form ?
1
2
7
1
1
1
1
2 a .
1
2 4 b . c . 1 d .
1 − −
1
1
1
Exercises
3. Reduced matrices below to reduced row echelon form
1
1
3
3
2
a .
1
2
4
− − b .
1
3
2
2
2
10
2
1
1
1
1
1
1
1
2
3 − − −
c .
2
2
1
1 2 d .
2
4
6
3
3
1
3
6
9