HW10-solutions - huang(dh34953 HW10 gilbert(54160 This...

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huang (dh34953) – HW10 – gilbert – (54160) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Four linearly independent vectors in R 4 span R 4 . True or False? 1. TRUE correct 2. FALSE Explanation: The space R 4 is 4-dimensional, so any four linearly independent vectors in R 4 will form a basis for R 4 , hence span R 4 . Consequently, the statement is TRUE . 002 10.0points The set H of all polynomials p ( x ) = a + bx 4 , a, b in R , is a subspace of the vector space P 6 of all polynomials of degree at most 6. True or False? 1. FALSE 2. TRUE correct Explanation: The zero polynomial p ( x ) = 0+0 x 4 belongs to H . So we need to check if the linear combination c 1 p 1 + c 2 p 2 of elements p 1 ( x ) = a 1 + b 1 x 4 , p 2 ( x ) = a 2 + b 2 x 4 in H also is a polynomial in H . But ( c 1 p 1 + c 2 p 2 )( x ) = c 1 p 1 ( x ) + c 2 p 2 ( x ) = c 1 ( a 1 + b 1 x 4 ) + c 2 ( a 2 + b 2 x 4 ) = ( c 1 a 1 + c 2 a 2 ) + ( c 1 b 1 + c 2 b 2 ) x 4 . Since c 1 a 1 + c 2 a 2 , c 1 b 1 + c 2 b 2 are in R , the linear combination c 1 p 1 + c 2 p 2 belongs to H . Consequently, the statement is TRUE . 003 10.0points The set H = a + 2 b a - b 3 b : a, b in R is a vector space under the usual addition and scalar multiplication of vectors in R 3 . True or False? 1. FALSE 2. TRUE correct Explanation: By matrix algebra, a + 2 b a - b 3 b = a 1 1 0 + b 2 - 1 3 , with a, b in R . Thus H consists of all linear combinations of the vectors u =
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