201 Quiz 3 TH solutions

# Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 3rd Quiz Solutions (Thursday) March 24, 2005 Notation. P n = space of polynomials, with real coefficients, of degree at most n . R m × n = space of m by n real matrices. Problem 1 Determine whether the following spaces are isomorphic. In case they are isomorphic, define an isomorphism relating them. Justify your answer. Solution Spaces are isomorphic if they have the same dimension. 1. R 2 and R 4 . No. 2. P 5 and R 5 No. 3. R 2 × 3 and R 6 Yes, under the natural identification. 4. P 5 and R 2 × 3 Yes, under the natural identification. 5. R 2 × k and C k , for k N . Yes, under the natural identification. Problem 2 Let V = C 1 ([0 , 1]) be the set of continuously differentiable func- tions on the closed interval [0 , 1]. V is a real linear space with respect to the operations of pointwise addition of functions and scalar multiplication. (a) To prove that the functions f ( x ) = cos x , g ( x ) = 2 x , and h ( x ) = e x are linearly independent in V , consider a relation c 1 cos x + c 2 2 x + c 3 e x = 0 . Obviously this can only be true if c 1 = c 2 = c 3

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