201Quiz3FR

# Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 3rd Quiz March 25, 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 Find out which of the following transformations are linear and for those that are linear, determine whether they are isomorphisms. 1. T : P 2 R , T ( f ( t )) = f (0). 2. T : C C , T ( x + iy ) = x - iy . 3. T : R 2 × 2 R 2 × 2 , T ( M ) = M 2 . Problem 2 Consider the (standard) basis B 1 = { 1 , x, x 2 , x 3 , x 4 } of P 4 . 1. Prove that the set B 2 = { x 4 , 2 x 3 , 1 - x 2 , 3 x - 1 , 2 x } is a basis of P 4 . Find the change of basis matrix
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Unformatted text preview: S from B 1 to B 2 . 2. Let T : P 4 → P 4 be the linear transformation deﬁned by T ( p ( x )) = p 00 ( x ) + p ( x ) + p ( x ) . Find the matrix of T with respect to B 2 . Problem 3 Given the subspace of R 2 × 2 S = { • x y z ‚ | £ 1-4 3 / • x y z ‚• 2-3 ‚ = 0 } , ﬁnd its dimension and a basis B of S such that [ A ] B = • 2 3 ‚ , for A = • 2-1 ‚ . 1...
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