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Unformatted text preview: MATH 235/W08 Assignment 4A Eigenvalues, Eigenvectors, Diagonalization Hand in questions 2,3,4,5,6,7,12. Due by 9:30 am on Wed. Feb. 27/08. 1. Find the eigenvalues and eigenvectors of A = 1 1 4 1 a 2 c b 2 . Is A diagonalizable? 2. Diagonalize, if possible, the following matrices, i.e. find the eigenvalues and then find the corresponding eigenspaces. A =  1 3 5 3 1 1 2 , B =  2 4 i 4 i 4 i 2 4 i 2 , C = 4 4 2 3 2 1 2 2 2 6 12 11 2 4 9 20 10 10 6 15 28 14 5 3 . 3. 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 for T relative to the bases { 1 ,t,t 2 } and { 1 ,t,t 2 ,t 3 ,t 4 } . 4. Suppose that the matrices A,B are similar. Prove that they have the same rank. 5. A 3 × 3 real matrix A has eigenvalues λ 1 = λ 2 = 2 ,λ 3 = 3 and associated eigenvectors v 1 = 1 1 , v 2 = 1 1 , v 3 = 1 1 1 , respectively. Find A . 1 6. Consider the matrix: A = 2 k 2 k 2 k where k ∈ R is a constant....
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This note was uploaded on 04/27/2010 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.
 Winter '08
 CELMIN
 Linear Algebra, Algebra, Eigenvectors, Vectors

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