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hw2solutions

# hw2solutions - Math 215 HW#2 Solutions 1 Problem 1.4.6...

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Math 215 HW #2 Solutions 1. Problem 1.4.6. Write down the 2 by 2 matrices A and B that have entries a ij = i + j and b ij = ( - 1) i + j . Multiply them to find AB and BA . Solution: Since a ij indicates the entry in A which is in the i th row and in the j th column, we see that A = 2 3 3 4 . Likewise, B = 1 - 1 - 1 1 . Therefore, AB = 2 3 3 4 1 - 1 - 1 1 = 2 · 1 + 3 · ( - 1) 2 · ( - 1) + 3 · 1 3 · 1 + 4 · ( - 1) 3 · ( - 1) + 4 · 1 = - 1 1 - 1 1 . Also, BA = 1 - 1 - 1 1 2 3 3 4 = 1 · 2 + ( - 1) · 3 1 · 3 + ( - 1) · 4 ( - 1) · 2 + 1 · 3 ( - 1) · 3 + 1 · 4 = - 1 - 1 1 1 . 2. Problem 1.4.16. Let x be the column vector (1 , 0 , . . . , 0). Show that the rule ( AB ) x = A ( Bx ) forces the first column of AB to equal A times the first column of B . Solution: Suppose that x has n components. Then, in order for Bx to make sense, B must be an m × n matrix for some m . In turn, for the matrix product AB to make sense, A must be an × m matrix for some . Now, suppose A = a 11 · · · a 1 m . . . . . . a 1 · · · a m and B = b 11 · · · b 1 n . . . . . . b m 1 · · · b mn . Then AB = a 11 · · · a 1 m . . . . . . a 1 · · · a m b 11 · · · b 1 n . . . . . . b m 1 · · · b mn = m i =1 a 1 i b i 1 ] · · · m i =1 a 1 i b in . . . . . . m i =1 a i b i 1 · · · m i =1 a i b in . Hence, ( AB ) x = m i =1 a 1 i b i 1 ] · · · m i =1 a 1 i b in . . . . . . m i =1 a i b i 1 · · · m i =1 a i b in 1 0 . . . 0 = m i =1 a 1 i b i 1 m i =1 a 2 i b i 1 . . . m i =1 a i b i 1 , 1

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which is just a copy of the first column of AB . On the other hand, Bx = b 11 · · · b 1 n . . . . . . b m 1 · · · b mn 1 0 . . . 0 = b 11 b 21 . . . b m 1 , which is the first column of B . Therefore, A ( Bx ) is A times the first column of B ; since A ( Bx ) = ( AB ) x and ( AB ) x is the first column of AB , we see that the first column of AB must be A times the first column of B . Though it’s not part of the assigned problem, the same argument with different choices of x (e.g. (0 , 1 , 0 , . . . , 0), etc.) will demonstrate that each column of AB must be equal to A times the corresponding column in B . 3. Problem 1.4.20. The matrix that rotates the xy -plane by an angle θ is A ( θ ) = cos θ - sin θ sin θ cos θ . Verify that A ( θ 1 ) A ( θ 2 ) = A ( θ 1 + θ 2 ) from the identities for cos( θ 1 + θ 2 ) and sin( θ 1 + θ 2 ). What is A ( θ ) times A ( - θ )? Solution: Using the definition of A ( θ 1 ) and A ( θ 2 ), we have that A ( θ 1 ) A ( θ 2 ) = cos θ 1 - sin θ 1 sin θ 1 cos θ 1 cos θ 2 - sin θ 2 sin θ 2 cos θ 2 = cos θ 1 cos θ 2 - sin θ 1 sin θ 2 - cos θ 1 sin θ 2 - sin θ 1 cos θ 2 sin θ 1 cos θ 2 + cos θ 1 sin θ 2 - sin θ 1 sin θ 2 + cos θ 1 cos θ 2 = cos( θ 1 + θ 2 ) - sin( θ 1 + θ 2 ) sin( θ 1 + θ 2 ) cos( θ 1 + θ 2 ) = A ( θ 1 + θ 2 ) , where I went from the second to the third lines using the identities for cos( θ 1 + θ 2 ) and sin( θ 1 + θ 2 ). Geometrically, the fact that A ( θ 1 ) A ( θ 2 ) = A ( θ 1 + θ 1 ) corresponds to the fact that rotating something by an angle of θ 2 and then rotating the result by an angle of θ 1 is
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hw2solutions - Math 215 HW#2 Solutions 1 Problem 1.4.6...

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