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asst4 - MATH 235 Fall 2007 Assignment#4 1 Let T P2 P4 be...

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MATH 235, Fall 2007 Assignment #4 1. Let T : P 2 P 4 be the transformation that maps a polynomial p ( t ) into the polynomial p ( t ) + t 2 p ( t ). (a) Find the image of p ( t ) = 2 - t + t 2 . (b) Show that T is a linear transformation. (c) Find the matrix T relative to the bases { 1 , t, t 2 } and { 1 , t, t 2 , t 3 , t 4 } . 2. Let A = 5 - 3 - 7 1 , and define T : R 2 R 2 by T ( x ) = A x . If possible, find a basis B for R 2 with the property that [ T ] B is diagonal, and state this diagonal matrix 3. (Proof of Theorem 9 from Section 5.5 of the text) Let A be a real 2 × 2 matrix with complex eigenvalue λ = a - bi ( b 6 = 0) and an associated eigenvector v in C 2 . (a) Prove that Re ( A v ) = A ( Re v ) and Im ( A v ) = A ( Im v ) . [Hint: Start by writing v =Re v + i (Im v ).] (b) Show that A ( Re v ) = aRe v + bIm v and A ( Im v ) = - bRe v + aIm v . [Hint: Write v = Re v + i Im v , and compute A v ]. (c) Let P = [ Re v Im v ] and C = a - b b a . Verify that AP = PC . (d) Prove that Re v and Im v are linearly independent. [Hint: Use one identity from each of (a) and (b), and prove by contradiction.] (e) Deduce that A = PCP - 1 .
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