Transcription of Linear Algebra: Graduate Level Problems and Solutions
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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.
Linear Algebra Igor Yanovsky, 2005 4 1 Basic Theory 1.1 Linear Maps Lemma. If A 2 Matmxn(F) and B 2 Matnxm(F), then tr(AB) = tr(BA): Proof. Note that the (i;i) entry in AB isPn P j=1 fiijflji, while (j;j) entry in BA is m i=1 fljifiij. Thus tr(AB) = Xm i=1 Xn j=1 fiijflji; tr(BA) = Xn j=1 Xm i=1 fljifiij: 1.2 Linear Maps as Matrices
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