{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

115a-4exam2sol - MATH 115A Lecture 4 Fall 2008 Midterm 2...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 2
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 T , then λ 2 is an eigenvalue of T 2 .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}