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# HW2_sol - 14 CHAPTER 1 Conversely suppose there exists a...

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14 CHAPTER 1 Conversely suppose there exists a nonsingular matrix M such that B = MA . Since M is nonsingular it is row equivalent to I . Thus there exist elementary matrices E 1 , E 2 , . . ., E k such that M = E k E k - 1 · · · E 1 I It follows that B = MA = E k E k - 1 · · · E 1 A Therefore B is row equivalent to A . 27. (a) The system V c = y is given by 1 x 1 x 2 1 · · · x n 1 1 x 2 x 2 2 · · · x n 2 . . . 1 x n +1 x 2 n +1 · · · x n n +1 c 1 c 2 . . . c n +1 = y 1 y 2 . . . y n +1 Comparing the i th row of each side, we have c 1 + c 2 x i + · · · + c n +1 x n i = y i Thus p ( x i ) = y i i = 1 , 2 , . . ., n + 1 (b) If x 1 , x 2 , . . ., x n +1 are distinct and V c = 0 , then we can apply part (a) with y = 0 . Thus if p ( x ) = c 1 + c 2 x + · · · + c n +1 x n , then p ( x i ) = 0 i = 1 , 2 , . . ., n + 1 The polynomial p ( x ) has n + 1 roots. Since the degree of p ( x ) is less than n + 1, p ( x ) must be the zero polynomial. Hence c 1 = c 2 = · · · = c n +1 = 0 Since the system V c = 0 has only the trivial solution, the matrix V must be nonsingular. SECTION 5 2. B = A T A = a T 1 a T 2 . . . a T n ( a 1 , a 2 , . . ., a n ) = a T 1 a 1 a T 1 a 2 · · · a T 1 a n a T 2 a 1 a T 2 a 2 · · · a T 2 a n . . . a T n a 1 a T n a 2 · · · a T n a n 5. (a) 1 1 1 2 1 2 4 - 2 1 2 3 1 1 1 2 + - 1 - 1 (1 2 3) = 6 0 1 11 - 1 4 (c) Let A 11 = 3 5 - 4 5 4 5 3 5 A 12 = 0 0 0 0 A 21 = (0 0) A 22 = (1 0)

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Section 5 15 The block multiplication is performed as follows: A 11 A 12 A 21 A 22 A T 11 A T 21 A T 12 A T 22 = A 11 A T 11 + A 12 A T 12 A 11 A T 21 + A 12 A T 22 A 21 A T 11 + A 22 A T 12 A 21 A T 21 + A 22 A T 22 = 1 0 0 0 1 0 0 0 0 6. (a) XY T = x 1 y T 1 + x 2 y T 2 + x 3 y T 3 = 2 4 1 2 + 1 2 2 3 + 5 3 4 1 = 2 4 4 8 + 2 3 4 6 + 20 5 12 3 (b) Since y i x T i = ( x i y T i ) T for j = 1 , 2 , 3, the outer product expansion of Y X T is just the transpose of the outer product expansion of XY T . Thus Y X T = y 1 x T 1 + y 2 x T 2 + y 3 x T 3 = 2 4 4 8 + 2 4 3 6 + 20 12 5 3 7. It is possible to perform both block multiplications. To see this suppose A 11 is a k × r
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HW2_sol - 14 CHAPTER 1 Conversely suppose there exists a...

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