Unformatted text preview: 8 Let F be a commuting family of matrices in C 3 × 3 . How many linearly independent matrices can F contain? What about the n × n case? 8 Let T be a linear operator on an n-dimensional space and suppose that T has n distinct characteristic values. Prove that any linear operator which commutes with T is a polynomial in T . 8 For V the vectorspace of n × n matrices over a field F , let the linear operator T A be defined for any fixed diagonal matrix A ∈ V by T A ( M ) = AM- MA . Show the family F of all linear operators T A is simultaneously diagonalizable. 8 Let V be a finite-dimensional vectorspace and let W 1 be any subspace of V . Prove that there is a subspace W 2 of V such that V = W 1 ⊕ W 2 . 8 If V = W 1 + ··· + W k and dim V = dim W 1 + ··· + dim W k then show the sum is direct. 8 Find a projection E such that E (1 ,- 1) = (1 ,- 1) and E (1 , 2) = (0 , 0). 8 If E 1 and E 2 are projections onto independent subspaces, then is E 1 + E 2 a projection?...
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This document was uploaded on 01/25/2012.
- Spring '09