Unformatted text preview: x âˆˆ V we have M x âˆˆ V ). d) Show that V Î» j is an invariant subspace for M . e) Show that V Î» j âˆ© V Î» k = { } if Î» j 6 = Î» k . f) Denote by Î» 1 ,...,Î» k the distinct eigenvalues of M (in other words, { Î» 1 ,...,Î» k } = { Î» 1 ,...,Î» n } ). Show that M is diagonalizable in F n if and only if V Î» 1 âŠ• ... âŠ• V Î» k = F n and explain why the sum before is a direct sum. 2. Find the eigenspaces of the matrix A = Â± 2 1 0 2 Â² Is the matrix diagonalizable? 1...
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 Winter '08
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 Math, Linear Algebra, Algebra

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