More Related Content Similar to 0912 ch 9 day 12 (20) More from festivalelmo (20) 0912 ch 9 day 121. 9.5 The Algebra of Matrices
Day Two
Ephesians 4:32 "Be kind to one another, tenderhearted,
forgiving one another, as God in Christ forgave you."
3. Matrix Multiplication
When doing A ⋅ B ,
# columns in A must = # rows in B
Inner Product of Row A and Column B
⎡ b1 ⎤
⎢ ⎥
b2 ⎥
⎡ a1 a2 → an ⎤ ⋅ ⎢
⎣ ⎦ ⎢ ⎥
↓ ⎥
⎢
⎢ bn ⎥
⎣ ⎦
has inner product of
a1b1 + a2b2 + ... + anbn
4. Matrix Multiplication
If A is an m x n matrix and
B is an n x k matrix
Then AB is the m x k matrix, C where cij
is the inner product of the
ith row of A and the jth column of B
5. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
6. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
7. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
8. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
dimension of product
9. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
dimension of product
AB11 : 5 ⋅ 5 + 9 ⋅ −1+ 2 ⋅1 → 18
10. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
dimension of product
AB11 : 5 ⋅ 5 + 9 ⋅ −1+ 2 ⋅1 → 18
AB21 : 6 ⋅ 5 + 5 ⋅ −1+ 3⋅1 → 28
11. ⎡ 5 ⎤
⎡ 5 9 2 ⎤ ⎢ ⎥
A = ⎢ ⎥ B = ⎢ −1 ⎥ Find AB
⎣ 6 5 3 ⎦ ⎢ 1 ⎥
⎣ ⎦
2x3 times 3x1
equal: we can multiply
dimension of product
AB11 : 5 ⋅ 5 + 9 ⋅ −1+ 2 ⋅1 → 18
AB21 : 6 ⋅ 5 + 5 ⋅ −1+ 3⋅1 → 28
⎡ 18 ⎤
AB = ⎢ ⎥
⎣ 28 ⎦
verify with calculator
12. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
13. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
14. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
15. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
16. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6
17. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
18. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
19. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
AB 32 : − 4 ⋅ 5 + 2 ⋅ −2 + 7 ⋅ −1 → − 31
20. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
⎡ 1 5 ⎤
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6 ⎢ ⎥
AB = ⎢ 6 10 ⎥
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
⎢ −2 −31 ⎥
⎣ ⎦
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
AB 32 : − 4 ⋅ 5 + 2 ⋅ −2 + 7 ⋅ −1 → − 31
21. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
⎡ 1 5 ⎤
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6 ⎢ ⎥
AB = ⎢ 6 10 ⎥
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
⎢ −2 −31 ⎥
⎣ ⎦
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
AB 32 : − 4 ⋅ 5 + 2 ⋅ −2 + 7 ⋅ −1 → − 31
Verify with calculator. Then do BA. Does AB=BA?
22. ⎡ 2 −1 7 ⎤ ⎡ 2 5 ⎤
⎢ ⎥ ⎢ ⎥
A = ⎢ 3 0 5 ⎥ B = ⎢ 3 −2 ⎥ Find AB
⎢ −4 2 7 ⎥
⎣ ⎦ ⎢ 0 −1 ⎥
⎣ ⎦
3x3 times 3x2 product : 3x2
AB11 : 2 ⋅ 2 + −1⋅ 3 + 7 ⋅ 0 → 1
AB12 : 2 ⋅ 5 + −1⋅ −2 + 7 ⋅ −1 → 5
⎡ 1 5 ⎤
AB 21 : 3⋅ 2 + 0 ⋅ 3 + 5 ⋅ 0 → 6 ⎢ ⎥
AB = ⎢ 6 10 ⎥
AB 22 : 3⋅ 5 + 0 ⋅ −2 + 5 ⋅ −1 → 10
⎢ −2 −31 ⎥
⎣ ⎦
AB 31 : − 4 ⋅ 2 + 2 ⋅ 3 + 7 ⋅ 0 → − 2
AB 32 : − 4 ⋅ 5 + 2 ⋅ −2 + 7 ⋅ −1 → − 31
Verify with calculator. Then do BA. Does AB=BA?
Nope. Matrix Multiplication is NOT commutative ...
25. Properties for Matrix Multiplication
A ( BC ) = ( AB ) C Associative
A ( B + C ) = AB + AC Distributive
26. Properties for Matrix Multiplication
A ( BC ) = ( AB ) C Associative
A ( B + C ) = AB + AC Distributive
( B + C ) A = BA + CA Distributive
but not = AB + AC
28. A Linear System written as a Matrix Equation
⎡ 1 1 1 ⎤ ⎡ x ⎤ ⎡ 1 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 2 −3 0 ⎥ ⎢ y ⎥ = ⎢ 2 ⎥
⎢ 2 6 2 ⎥ ⎢ z ⎥ ⎢ 5 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
29. A Linear System written as a Matrix Equation
⎡ 1 1 1 ⎤ ⎡ x ⎤ ⎡ 1 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 2 −3 0 ⎥ ⎢ y ⎥ = ⎢ 2 ⎥
⎢ 2 6 2 ⎥ ⎢ z ⎥ ⎢ 5 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎧ x + y + z = 1
⎪
⎨2x − 3y = 2
⎪2x + 6y + 2z = 5
⎩
30. A Linear System written as a Matrix Equation
⎡ 1 1 1 ⎤ ⎡ x ⎤ ⎡ 1 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 2 −3 0 ⎥ ⎢ y ⎥ = ⎢ 2 ⎥
⎢ 2 6 2 ⎥ ⎢ z ⎥ ⎢ 5 ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎧ x + y + z = 1
⎪
⎨2x − 3y = 2
⎪2x + 6y + 2z = 5
⎩
Read Example 7 on pp. 682, 683
Read Computer Graphics on pp. 683, 684
31. HW #11
Let us always meet each other with smile, for the smile
is the beginning of love.
Mother Teresa
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