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Linear Algebra: Graduate Level Problems and Solutions

Linear algebra : Graduate Level Problems and SolutionsIgor Yanovsky1 Linear AlgebraIgor Yanovsky, 20052 Disclaimer:This handbook is intended to assist Graduate students with qualifyingexamination preparation. Please be aware, however, that the handbook might contain,and almost certainly contains, typos as well as incorrect or inaccurate Solutions . I cannot be made responsible for any inaccuracies contained in this AlgebraIgor Yanovsky, 20053 Contents1 Basic Linear Maps .. Linear Maps as Matrices .. Dimension and Isomorphism .. Matrix Representations Redux .. Subspaces .. Linear Maps and Subspaces .. Dimension Formula .. Matrix Calculations .. Diagonalizability ..82 Inner Product Inner Products .. Orthonormal Bases .. Gram-Schmidt procedure .. QR Factorization .. Orthogonal Complements and Projections ..93 Linear Maps on Inner Product Adjoint Maps .. Self-Adjoint Maps .. Polarization and Isometries .. Unitary and Orthogonal Operators.

Linear Algebra Igor Yanovsky, 2005 8 1.9 Diagonalizability Deflnition. Let T be a linear operator on the flnite-dimensional space V. T is diagonalizable if there is a basis for V consisting of eigenvectors of T. Theorem. Let v1;:::;vn be nonzero eigenvectors of distinct eigenvalues ...

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