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# hw6 - they’re not the exact same subspace Use this to...

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HOMEWORK 6 1. Diagonalize the following matrix, or show that it’s not diagonalizable (a) over R ; and (b) over C : A = 1 - 1 1 1 1 1 0 0 1 2. Deﬁne T : P 2 ( C ) P 2 ( C ) by: T ( ax 2 + bx + c ) = 3 ax 2 + (2 b + a ) x + ( c + 2 a ) Diagonalize T , or show that it’s not diagonalizable. 3. Deﬁne T : P ( C ) P ( C ) by T ( x n ) = x n +( - 1) n and extend by linearity. Diagonalize T or show that it’s not diagonalizable. 4. Let A be a square matrix. (a) Show that A and A T have the same characteristic polynomial. (b) As a consequence of (a), A and A T have the same eigenvalues. Suppose λ is an eigenvalue of A . Show by example that the eigenspace for A corresponding to λ need not be the same as the eigenspace for A T corresponding to λ . (c) Despite (b), the two eigenspaces will always have the same dimensions even if
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Unformatted text preview: they’re not the exact same subspace. Use this to show that A is diagonalizable iﬀ A T is too. 5. Let A be a square invertible matrix. (a) Show that if A has an eigenvalue λ , then A-1 has λ-1 as an eigenvalue, and that the eigenvectors of A corresponding to λ are precisely the eigenectors of A-1 corresponding to λ-1 . (b) Show that A is diagonalizable iﬀ A-1 is too. 6. Suppose V is a real vector space and T ∈ L ( V,V ) has no (real) eigenvalues. Prove that every T-invariant subspace of V has even dimension....
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