# asmt2 - A as A B = a-b b a c-d d c = a c b d b d a c and...

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McGill University Math 270-2010 Fall: Applied Linear Algebra Written Assignment 2: due on Monday, Oct. 18, 2010 1. Vector Space (a) In vector space P 2 , let ( S ) be the set consisting of three quadratics: 1 + 2 x + 3 x 2 , 4 + 5 x + 6 x 2 , 7 + 8 x + 9 x 2 . Does 1 + x + x 2 belong to span( S )? Which quadratics a + bx + cx 2 are in span( S )? (b) Assume that in a vector space V , the vector u and v are linearly independent. Prove that the set { 2 u - v , u + 5 v } is linearly independent. (c) In vector space P 2 , take the basis 1 , x, x 2 - 1 2 . What is the coordinate vector of 3 x 2 - 5 x ? (d) If f is a polynomial of degree n , show that f, f , · · · , f n form a basis of P n . (e) Show that the vectors (1 , 2 , 3) T , ( - 1 , 0 , 1) T , (4 , 9 , 7) T for a basis for R 3 . (f) Given a collection A of real matrices of the form A = [ a - b b a ] , define the addition in
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Unformatted text preview: A as A + B = [ a-b b a ] + [ c-d d c ] = [ a + c-( b + d ) ( b + d ) a + c ] and the scalar multiplication as α [ a-b b a ] = [ αa-αb αb αa ] . • Show that the collection ( A ) forms a vector space. • Find a basis of ( A ) and give dim( A ). 2. Inner Product Space (a) Let x = α 1 e 1 + α 2 e 2 and y = β 1 e 1 + β 2 e 2 are a pair of geometric vectors in R 2 . Verify that one may deﬁne the following inner product in R 2 . ( x , y ) = 2 α 1 β 1 + 3 α 2 β 2 . (b) If y 1 , · · · , y n are vectors orthogonal to x , show that any vector in span( y 1 , · · · , y n ) is orthogonal to x ....
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