{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

prelim2_09Sol

# prelim2_09Sol - Prelim 2 Solutions Math 2940 1 Let 0 2 2 A...

This preview shows pages 1–3. Sign up to view the full content.

Prelim 2 Solutions Math 2940 November 19, 2009 1. Let A = 0 - 2 2 - 2 0 2 - 2 - 2 4 . (a) (10 points) Determine the eigenvalues of A , and a basis for R 3 consisting of eigenvectors of A . Solution : f A ( λ ) = det ( A - λI 3 ) = - λ 3 + 4 λ 2 - 4 λ = - λ ( λ - 2) 2 = 0 -→ λ = 0 , 2 , 2 λ = 0 : E 0 = ker 0 - 2 2 - 2 0 2 - 2 - 2 4 = span 1 1 1 λ = 2 : E 2 = ker - 2 - 2 2 - 2 - 2 2 - 2 - 2 2 = span 1 - 1 0 , 1 0 1 eigenbasis : 1 1 1 , 1 - 1 0 , 1 0 1 (b) (10 points) Consider the discrete dynamical system ~x ( t + 1) = A ~x ( t ) , ~ x 0 = ~x (0) = 2 1 2 . Using part (a), find a formula for ~x ( t ), for t 1. Solution : x 0 = 2 1 2 = C 1 1 1 1 + C 2 1 - 1 0 + C 3 1 0 1 so C 1 = 1 , C 2 = 0 , C 3 = 1 ~x ( t ) = C 1 λ t 1 ~ v 1 + C 2 λ t 2 ~ v 2 + C 3 λ t 3 ~ v 3 = 0 t ~ v 1 + 2 t ~ v 3 = 0 t 1 1 1 + 2 t 1 0 1 = 2 t 1 0 1

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

View Full Document
(c) (5 points) Determine an invertible matrix S and a diagonal matrix D such that D = S - 1 AS . (You do not need to compute S - 1 !). Solution : D = λ 1 0 0 0 λ 2 0 0 0 λ 3 = 0 0 0 0 2 0 0 0 2 S = ~ v 1 ~ v 2 ~ v 3 = 1 - 1 1 1 1 0 1 0 1 2. Let V = span { e t , te t , t 2 e t } . You are told that these three functions form a basis B of V . Let T : V -→ V be given by T ( f ) = f 0 ( t ) - f ( t ) . (a) (7 points) Find a basis for the kernel of T . (b) (7 points) Find a basis for the image of T . (c) (7 points) Determine the matrix of T with respect to the basis B .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

prelim2_09Sol - Prelim 2 Solutions Math 2940 1 Let 0 2 2 A...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online