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Unformatted text preview: CS 113 Linear Systems of Equations Linear Algebraic Equations Nonlinear Equations: Example Representing Linear Algebraic Equations
in Matrix Form Ax = b Representing Linear Algebraic Equations
in Matrix Form Ax = b
Ⱥ a11
Ⱥ
Ⱥa21
A=
Ⱥ M
Ⱥ
Ⱥan1 a12
a22
M
an 2 L a1n Ⱥ
Ⱥ
L a2 n Ⱥ
O M Ⱥ
Ⱥ
L ann Ⱥ “coefficient matrix” € € Ⱥ x1 Ⱥ
Ⱥ
Ⱥ
x 2 Ⱥ
Ⱥ
x = Ⱥ M Ⱥ
Ⱥ
Ⱥ
x n −1 Ⱥ
Ⱥ
Ⱥ x n Ⱥ € “vector of unknowns” € Ⱥ b1 Ⱥ
Ⱥ Ⱥ
Ⱥ b2 Ⱥ
b = Ⱥ M Ⱥ
Ⱥ Ⱥ
Ⱥbn −1 Ⱥ
Ⱥ bn Ⱥ
“vector of known constants” Representing Linear Algebraic Equations
in Matrix Form: EXAMPLE #1 Ax = b
Ⱥ 5 −3
7 Ⱥ
Ⱥ
Ⱥ
A = Ⱥ2.1 0.7
0 Ⱥ
Ⱥ 0 3.1 −7.2 Ⱥ
Ⱥ
Ⱥ
“coefficient matrix” € € Ⱥ x1 Ⱥ
Ⱥ Ⱥ
x = Ⱥ x 2 Ⱥ
Ⱥ x 3 Ⱥ
Ⱥ Ⱥ Ⱥ −1 Ⱥ
Ⱥ Ⱥ
b = Ⱥ1.5 Ⱥ
Ⱥ8.1Ⱥ
Ⱥ Ⱥ € “vector of unknowns” € “vector of known constants” Example #2: Linear Algebraic Equation Ⱥ 3 2 Ⱥ
A = Ⱥ
Ⱥ
Ⱥ−1 2 Ⱥ Ⱥ18 Ⱥ
b = Ⱥ Ⱥ
Ⱥ 2 Ⱥ
€ € Ⱥ x1 Ⱥ
x = Ⱥ Ⱥ
Ⱥ x 2 Ⱥ Ax = b
The Graphic Method Review of Matrix Notation
1. Matrix
2. Row vectors and column vectors
3. Square matrices, principal or main diagonal
4. Special types of square matrices: symmetric, diagonal,
identity, upper triangular, lower triangular
5. Matrix operating rules: addition and multiplication MATRICES ORDER or SIZE of MATRICES Order: 4 ×3 SPECIFYING ELEMENTS A= Ⱥ  5
Ⱥ
2
Ⱥ
Ⱥ 9
Ⱥ
Ⱥ 3 0 1 Ⱥ
 4Ⱥ
3
Ⱥ
2 6 Ⱥ
Ⱥ
1 4 Ⱥ a1 2 = 0
a2 1 = 2 a4 1 = 3 a 2 3 = 4 a4 3 = 4 a3 2 = 2 TRANSPOSE OF A MATRIX Ⱥ  5
Ⱥ
Ⱥ 2
Ⱥ 9
Ⱥ
Ⱥ 3 0 1 Ⱥ
Ⱥ
3  4Ⱥ
2 6 Ⱥ
Ⱥ
1 4 Ⱥ = Ⱥ 5
Ⱥ
Ⱥ 0
Ⱥ
Ⱥ 1
Ⱥ 2
3 9
2 4 6 3 Ⱥ
Ⱥ
1 Ⱥ
Ⱥ
4 Ⱥ
Ⱥ IN MATLAB:
>> A=[5 0 1; 2 3 –4;9 2 6;3 1 4];
>> B=A’
the apostrophe denotes the transpose operation MULTIPLICATION BY A SCALAR
Ⱥ  5
Ⱥ
2
3 × Ⱥ
Ⱥ –9
Ⱥ
Ⱥ 3 0 1 Ⱥ
Ⱥ
3  4Ⱥ
2 6 Ⱥ
Ⱥ
1 4 Ⱥ = Ⱥ–15
Ⱥ
Ⱥ 6
Ⱥ 27
Ⱥ
Ⱥ 9 0
9
6
3 3 Ⱥ
Ⱥ
–12Ⱥ
18 Ⱥ
Ⱥ
12 Ⱥ IN MATLAB: >> A = [5 0 1; 2 3 –4; 9 2 6; 3 1 4];!
>> B = 3*A!
simply multiply by scalar ADDITION AND SUBTRACTION
Ⱥ  5
Ⱥ
Ⱥ 2
Ⱥ 9
Ⱥ
Ⱥ 3 0 1 Ⱥ
 4Ⱥ
3
Ⱥ
2 6 Ⱥ
Ⱥ
1 4 Ⱥ + Ⱥ 1
3
2 Ⱥ
Ⱥ 5Ⱥ
Ⱥ 1 4
Ⱥ
Ⱥ 3
1  2Ⱥ
Ⱥ
 2 0 Ⱥ
Ⱥ 4
Ⱥ = Ⱥ 4 3
3 Ⱥ
Ⱥ
 9Ⱥ
7
Ⱥ 1
Ⱥ
Ⱥ  6 3
4 Ⱥ
Ⱥ
 1 4 Ⱥ
Ⱥ 7
Ⱥ IN MATLAB:
>> A = [5 0 1; 2 3 –4; 9 2 6; 3 1 4];!
>> B = [1 3 2; 1 4 –5; 3 1 –2; 4 –2 0];!
>> C = A + B;! ADDITION AND SUBTRACTION Ⱥ2  1Ⱥ
Ⱥ 1 0 2Ⱥ Ⱥ
+ 2  1Ⱥ =
Ⱥ
Ⱥ
Ⱥ
Ⱥ 3 1 1 Ⱥ Ⱥ
Ⱥ2  1Ⱥ
Ⱥ
Ⱥ Matrices must have the same order (or size) MULTIPLICATION OF MATRICES: = 4 Rows
4 Rows 6 Columns 6 Columns The first matrix determines the number of rows
and the second determines the number of columns Example 1
3×2 Ⱥ 1 0 2 Ⱥ Ⱥ2
Ⱥ
Ⱥ Ⱥ3
Ⱥ 3 1 1 Ⱥ Ⱥ
Ⱥ1
Ⱥ 2×3 1 Ⱥ
Ⱥ = Ⱥ 0
2 Ⱥ
Ⱥ Ⱥ *
4 Ⱥ
Ⱥ 2×2 * Ⱥ
Ⱥ
* Ⱥ Example
3×2 Ⱥ 1 0 2 Ⱥ Ⱥ2
Ⱥ
Ⱥ Ⱥ3
Ⱥ 3 1 1 Ⱥ Ⱥ
Ⱥ1
Ⱥ 2×3 1 Ⱥ
Ⱥ = Ⱥ 0
2 Ⱥ
Ⱥ Ⱥ *
4 Ⱥ
Ⱥ 2×2 –7 Ⱥ
Ⱥ
* Ⱥ Example
3×2 Ⱥ 1 0 2 Ⱥ Ⱥ2
Ⱥ
Ⱥ Ⱥ3
Ⱥ 3 1 1 Ⱥ Ⱥ
Ⱥ1
Ⱥ 2×3 1 Ⱥ
Ⱥ = Ⱥ 0
2 Ⱥ
Ⱥ Ⱥ10
4 Ⱥ
Ⱥ 2×2 –7 Ⱥ
Ⱥ
* Ⱥ Example
3×2 Ⱥ 1 0 2 Ⱥ Ⱥ2
Ⱥ
Ⱥ Ⱥ3
Ⱥ 3 1 1 Ⱥ Ⱥ
Ⱥ1
Ⱥ 2×3 1 Ⱥ
Ⱥ = Ⱥ 0
2 Ⱥ
Ⱥ Ⱥ10
4 Ⱥ
Ⱥ 2×2 –7 Ⱥ
Ⱥ
–9 Ⱥ Example 2
3×3 Ⱥ 1
Ⱥ
2
Ⱥ
Ⱥ 4
Ⱥ 5
6
2 2×3 3 Ⱥ
Ⱥ Ⱥ 1 0 2 Ⱥ
8 Ⱥ Ⱥ
Ⱥ
Ⱥ 3 1 1 Ⱥ
1 Ⱥ
Ⱥ Must have enough elements to match up
# Cols in (i) must = # Rows in (ii) Example 3: Inner Product or Scalar Product or Dot Product
2×1
1×2 [4 1×1 Ⱥ1 Ⱥ
 2 ]Ⱥ Ⱥ = [ 6 ]
Ⱥ5 Ⱥ IN MATLAB: >> a = [4 2];!
>> b = [1; 5];!
>> c = a * b! MATRIX MULTIPLICATION IS NOT COMMUTATIVE Ⱥ2 0 Ⱥ Ⱥ1 2 Ⱥ Ⱥ2 4 Ⱥ
Ⱥ
Ⱥ Ⱥ
Ⱥ = Ⱥ
Ⱥ
0 1 Ⱥ Ⱥ0 1 Ⱥ Ⱥ0 1 Ⱥ
Ⱥ Ⱥ1 2 Ⱥ Ⱥ2 0 Ⱥ Ⱥ2 2 Ⱥ
Ⱥ
Ⱥ Ⱥ
Ⱥ = Ⱥ
Ⱥ
0 1 Ⱥ Ⱥ0 1 Ⱥ Ⱥ0 1 Ⱥ
Ⱥ In general, if A and B are two matrices, AB = B A IDENTITY MATRICES
These behave like 1 in ordinary arithmetic multiplication
e.g., 3×1=3 and 1×3=3
Identity matrices are always square and
have 1s on the diagonal and 0s
elsewhere
Ⱥ1 0 Ⱥ Ⱥ3 1 Ⱥ Ⱥ3 1 Ⱥ
Ⱥ
Ⱥ Ⱥ
Ⱥ = Ⱥ
Ⱥ
0 1 Ⱥ Ⱥ1 5 Ⱥ Ⱥ1 5 Ⱥ
Ⱥ Ⱥ3 1 Ⱥ Ⱥ1 0 Ⱥ Ⱥ3 1 Ⱥ
Ⱥ
Ⱥ Ⱥ
Ⱥ = Ⱥ
Ⱥ
1 5 Ⱥ Ⱥ0 1 Ⱥ Ⱥ1 5 Ⱥ
Ⱥ Multiplication by an identity matrix is commutative
We write I for an identity matrix IDENTITY MATRICES Identity matrices times a matrix or
vector produces the same matrix or
vector Ix = x € THE INVERSE OF A MATRIX
In ordinary arithmetic multiplication the inverse of 3 is
and the inverse of is 3 since For square matrices the inverse of a matrix A is written A–1 ,
when it exists.
A A–1 = I and A–1 A = I where I is the identity matrix.
Multiplication with an inverse matrix is commutative. FINDING THE INVERSE OF A MATRIX IN MATLAB
define a random matrix 5x5 use the command “inv” to find the inverse of A and and assign
it to the variable B (notice how B is also a 5x5 matrix) see how B*A = I (identity matrix) also A*B = I (identity matrix) USING THE INVERSE TO SOLVE LINEAR SYSTEM OF
EQUATIONS set up a matrix equation Ax = b
Find the inverse of A and multiply both sides of the equation by A–1 −1 −1 A Ax = A b €
€ −1 Ix = A b
−1 x=A b EXAMPLE #1
Ⱥ5 x1 − 3 x 2 + 7 x 3 = −1
Ⱥ Ⱥ 2.1x1 + 0.7 x 2 = 1.5
Ⱥ Ⱥ 3.1x 2 − 7.2 x 3 = 8.1 € € Ⱥ 5 −3
7 Ⱥ
Ⱥ
Ⱥ
A = Ⱥ2.1 0.7
0 Ⱥ
Ⱥ 0 3.1 −7.2 Ⱥ
Ⱥ
Ⱥ Ax = b
Ⱥ x1 Ⱥ
Ⱥ Ⱥ
x = Ⱥ x 2 Ⱥ
Ⱥ x 3 Ⱥ
Ⱥ Ⱥ € Ⱥ 0.2017 −0.004 0.1961 Ⱥ
Ⱥ
Ⱥ
€
A −1 = Ⱥ −0.6050 1.4406 −0.5882 Ⱥ
Ⱥ −0.2605 0.6202 −3.922 Ⱥ
Ⱥ
Ⱥ € Ⱥ −1 Ⱥ
Ⱥ Ⱥ
b = Ⱥ1.5 Ⱥ
Ⱥ8.1Ⱥ
Ⱥ Ⱥ Ⱥ x1 Ⱥ
Ⱥ 1.3806 Ⱥ
Ⱥ Ⱥ
Ⱥ
Ⱥ
€ x = Ⱥ x 2 Ⱥ = A −1b = Ⱥ −1.9988 Ⱥ
Ⱥ x 3 Ⱥ
Ⱥ −1.9856 Ⱥ
Ⱥ Ⱥ
Ⱥ
Ⱥ EXAMPLE #1: FROM MATLAB Ax = b € NOT USING THE INVERSE TO SOLVE LINEAR
SYSTEM OF EQUATIONS set up a matrix equation Ax = b
THERE ARE OTHER WAYS TO SOLVE THIS MATRIX
EQUATION THAT DO NOT RELY ON FINDING AN INVERSE. THESE
METHODS TEND TO BE MORE COMPUTATIONALLY EFFICIENT. € ONE POPULAR METHOD IS CALLED GAUSS ELIMINATION BUT
THERE ARE MANY OTHERS. MATLAB HAS THESE METHODS
BUILT IN. NOT USING THE INVERSE TO SOLVE LINEAR
SYSTEM OF EQUATIONS Ax = b
TO IMPLEMENT A NONINVERSE METHOD FOR
SOLVING A SYSTEM
OF EQUATIONS IN MATLAB
USE THE BACKSLASH “\” OR
FORWARD SLASH “/” € Noninverse method
of solving system of
equations ...
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 Fall '10
 PhillipRegali

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