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MA 405 Test 3 Study Guide

MA 405 Test 3 Study Guide - MA 405 Test 3 Study Guide The...

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MA 405 April 12, 2009 Test 3 - Study Guide The topics for Test 3 are covered in Sections 6.1 - 6.3, 5.1 - 5.5 in the textbook. Learning objectives: define and compute eigenvalues, eigenvectors, eigenspaces; recognize diagonalizable matrices; be able to write a diagonalizable matrix as A = PDP - 1 , for some invertible matrix P and diagonal matrix D ; use it to compute powers of A . apply the dot product, and use its properties; relationship between the dot product and matrix mul- tiplication; identify orthogonal subspaces; define the orthogonal complement of a vector space; define the four fundamental subspaces associated with a matrix; identify the relationship between them; compute the basis for each; use their properties; calculate the least-squares solution of an inconsistent linear system; compute the residual and the projection of a vector onto a subspace on R n ; calculate the equation of a line/parabola that best fits a given set of data; inner product spaces; definitions and properties; be able to check that an operation is an inner product; identify orthogonal/orthonormal sets of vectors in an inner product space; calculate the coordinates of a vector with respect to an orthogonal/orthonormal basis; identify orthogonal matrices and use their properties; 1
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Practice Questions 1. Consider the matrix A = 2 3 0 4 3 0 0 0 6 . (a) Find the eigenvalues of A . (b) Find bases for the eigenspaces corresponding to each eigenvalue of A . (c) Check that A is diagonalizable. Find the invertible matrix P that diagonalizes A and the diagonal matrix D which is similar to A, that is, P - 1 AP = D .
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