MATH 235 Final: MATH 235 UMass Amherst final-compressed

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turquoisegnu128

31 Jan 2019

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Fall 2000: (15 points) the matrices a and b below are row equivalent (you do not need to check this fact). Justify your answers: (15 points) (a) show that the characteristic polynomial of the matrix a = is ( 1)( + 1)( 2). 3 (b) find a basis of r3 consisting of eigenvectors of a. (c) find an invertible matrix p and a diagonal matrix d such that the matrix. P 1ap = d: (12 points) determine for which of the following matrices a below there exists an invertible matrix p (with real entries) such that p 1ap is a diagonal matrix. You do not need to nd p . 0 0 (cid:19) (c) (cid:18) 0 1: (20 points) let w be the plane in r3 spanned by v1 = . Note: parts 5a, 5b are mutually independent and are not needed for doing parts.

Related Questions

4. (12 points) Let the matrix A and its reduced row echelon form R be given by the following. A= [ 1 -2 2 -3 1 -2 4 -1 4 1 -3] [1001 1] -4 0 0 1 0 1 1 7 -1 0 0 1 1 -1 | = R. (a) Find a basis for the column space of A. (b) Find a basis for the row space of A. (c) Find a basis for the null space of A. (d) Verify that the dimension of the column space of A plus the dimension of the null space of A is equal to the number of columns of A.

10 please