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MAT 272 Assignment 5 - solutions

# MAT 272 Assignment 5 - solutions - MAT 272 Assignment#5 Due...

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MAT 272 Assignment #5 Due: October 27, 2009 1) Show that the set V of all 2×3 matrices is a vector space. Let i) ii) iii) iv) v) vi) vii)

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viii) ix) x) 2) Show that the set H of 2×3 matrices of the form is a subspace of the vector space V in #1. i) ii) iii) 3) Show that the set of polynomials of the form is a vector space. Let p(x) = a 2 x 2 +a 1 x+a 0 , q(x) = b 2 x 2 +b 1 x+b 0 , and r(x) = c 2 x 2 +c 1 x+c 0 . Further let k and l be scalars. i) p(x)+q(x) = a 2 x 2 +a 1 x+a 0 + b 2 x 2 +b 1 x+b 0 = (a 2 + b 2 )x 2 + (a 1 +b 1 )x + (a 0 +b 0 ) ii) p(x)+q(x) = a 2 x 2 +a 1 x+a 0 + b 2 x 2 +b 1 x+b 0 = (a 2 + b 2 )x 2 + (a 1 +b 1 )x + (a 0 +b 0 ) = (b 2 + a 2 )x 2 + (b 1 +a 1 )x + (b 0 +a 0 ) = a 2 x 2 +a 1 x+a 0 + b 2 x 2 +b 1 x+b 0 = q(x) + p(x). iii) (p(x)+q(x)) +r(x) = [(a 2 + b 2 )x 2 + (a 1 +b 1 )x + (a 0 +b 0 )] + c 2 x 2 +c 1 x+c 0 = [(a 2 + b 2 )+ c 2 ]x 2 + [(a 1 +b 1 )+ c 1 ]x + [(a 0 +b 0 ) c 0 ] = [a 2 + (b 2 + c 2 )]x 2 + [a 1 +(b 1 + c 1 )]x + [a 0 +(b 0 c 0 )] = p(x) +(q(x)+r(x)) iv) p(x)+0(x) = a 2 x 2 +a 1 x+a 0 + 0x 2 +0x+0 = (a 2 + 0)x 2 + (a 1 +0)x + (a 0 +0) = a 2 x 2 +a 1 x+a 0 = p(x) v) a 2 x 2 +a 1 x+a 0 + -a 2 x 2 +-a 1 x+-a 0 = 0x 2 +0x+0. So –p(x) = -a 2 x 2 +-a 1 x+-a 0 vi) kp(x) = k(a 2 x 2 +a 1 x+a 0 ) = ka 2 x 2 +ka 1 x+ka 0 vii) k(p(x)+q(x)) = k[(a 2 + b 2 )x 2 + (a 1 +b 1 )x + (a 0 +b 0 )] = (ka 2 + kb 2 )x 2
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