MAT 2401 Linear Algebra
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Transcript of MAT 2401 Linear Algebra
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MAT 2401Linear Algebra
2.1 Operations with Matrices
http://myhome.spu.edu/lauw
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HW...
If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE.
GJE is the corner stone of this class, you really need to figure it out.
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Today
Written HW Again, today may be longer. It is
more efficient to bundle together some materials from 2.2.
Next class session will be shorter.
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Preview
Look at the algebraic operations of matrices
“term-by-term” operations•Matrix Addition and Subtraction
•Scalar Multiplication Non-“term-by-term” operations
•Matrix Multiplication
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Matrix
If a matrix has m rows and n columns, then the size (dimension) of the matrix is said to be mxn.
1 2
1
2
n
m
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Notations
Matrix
th
t
h
ij
ij
j
A a
ai
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Notations
Matrix Example:
11
23
1 1 1 4
2 2 5 11
4 6 8 24
A
a
a
th
t
h
ij
ij
j
A a
ai
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Special Cases
Row Vector
Column Vector
1 2 nb b b
1
2
m
c
c
c
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Matrix Addition and Subtraction
Let A = [aij] and B = [bij] be mxn matrices
Sum: A + B = [aij+bij]
Difference: A-B = [aij-bij]
(Term-by term operations)
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Example 1
1 2
3 1
0 2
3 2
A
B
A B
A B
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Scalar Multiplication
Let A = [aij] be a mxn matrix and c a scalar.
Scalar Product: cA=[caij]
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Example 2
1 2
3 1A
2A
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Matrix Multiplication
Define multiplications between 2 matrices
Not “term-by-term” operations
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Motivation
2 3 4 5x y z
The LHS of the linear equation consists of two pieces of information:•coefficients: 2, -3, and 4
•variables: x, y, and z
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Motivation
2 3 4 5
2 3 4 5
x y z
x
y
z
Since both the coefficients and variables can be represented by vectors with the same “length”, it make sense to consider the LHS as a “product” of the corresponding vectors.
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Row-Column Product
1
21 2 1 1 2 2n n n
n
b
ba a a a b a b a b
b
same no. of elements
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Example 3
2
21 3 2 4
1
2
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Matrix Multiplication
1
21
11 12 111 1
21 22
2
2
1
1 1 2
1
j
ji i ip
pj
i j i j
pn
n
p pnm m mp
ip pj
b
ba a a
b
a b a
a a ab b
b b
b b
b a
a
b
a a
th
th ijc
j
i
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Example 4
1 2 0 1
1 0 1 0
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Example 5 (a)
4 21 2 1
0 12 3 1
2 1
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
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Example 5 (b)
1 2 3 2
2 3 1 3
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
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Example 5 (c)
11 2
1
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
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Example 5 (d)
11 2
1
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Remark: 11 2 ,
1A B
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Example 5 (e)
1 1 1 1
1 1 1 1
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Remark:1 1 1 1
, 1 1 1 1
A B
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Example 5 (f)
1 0 1 2
0 1 3 4
Scratch:Q: Is it possible to multiply the 2 matrices?
Q: What is the dimension of the resulting matrix?
Remark:1 0 1 2
, 0 1 3 4
I A
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Interesting Facts
The product of mxp and pxn matrices is a mxn matrix.
In general, AB and BA are not the same even if both products are defined.
AB=0 does not necessary imply A=0 or B=0.
Square matrix with 1 in the diagonal and 0 elsewhere behaves like multiplicative identity.
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Identity Matrix
nxn Square Matrix
1 0 0
0 1
0
0 0 1
nI I
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Zero Matrix
mxn Matrix with all zero entries
0 0 0
0 00 0
0
0 0 0
mn
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Representation of Linear System by Matrix Multiplication
4
2 2 5 11
4 6 8 24
x y z
x y z
x y z
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Representation of Linear System by Matrix Multiplication
2 2 5 11
4
4
4 6 8 2
x y z
z
x y z
x y
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Representation of Linear System by Matrix Multiplication
4
4 6 8
11
2
2 5
4
2x y
y
z
x z
x y z
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Let
Then the linear system is given by
Representation of Linear System by Matrix Multiplication
4
2 2 5 11
4 6 8 24
x y z
x y z
x y z
1 1 1 4
2 2 5 , , 11
4 6 8 24
x
A X y b
z
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Let
Then the linear system is given by
Remark
It would be nice if “division” can be defined such that:
(2.3) Inverse
1 1 1 4
2 2 5 , , 11
4 6 8 24
x
A X y b
z
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HW...
If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE.
GJE is the corner stone of this class, you really need to figure it out.