115a-4exam2sol

# 115a-4exam2sol - MATH 115A - Lecture 4 - Fall 2008 Midterm...

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MATH 115A - Lecture 4 - Fall 2008 Midterm 2 - November 14, 2008 NAME: STUDENT ID #: This is a closed-book and closed-note examination. Calculators are not allowed. Please show all your work. Use only the paper provided. You may write on the back if you need more space, but clearly indicate this on the front. There are 5 problems for a total of 100 points. POINTS: 1. 2. 3. 4. 5. 1

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2 1. (20 points) Let P 3 be the R -vector space of polynomials of degree at most 3. Let A = { 1 ,x,x 2 ,x 3 } be the standard ordered basis of P 3 and let T : P 3 → P 3 be the linear transformation such that T ( f )( x ) = f ( x ) + 2 f 0 ( x ) - f 00 ( x ). Compute the matrix representation [ T ] A . We calculate: T (1) = 1 T ( x ) = x + 2 T ( x 2 ) = x 2 + 4 x - 2 T ( x 3 ) = x 3 + 6 x 2 - 6 x to obtain the matrix representation [ T ] A = 1 2 - 2 0 0 1 4 - 6 0 0 1 6 0 0 0 1
3 2. (20 points) Suppose V is an F -vector space and T : V V is a linear transformation. Show: if λ F is an eigenvalue of

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## This note was uploaded on 02/21/2010 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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115a-4exam2sol - MATH 115A - Lecture 4 - Fall 2008 Midterm...

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