Unformatted text preview: A is given by t 2 ( t 21) 3 ( t2) 2 . (a) What is the value of n . (b) Compute the trace and determinant of A . (c) Find the eigenvalues of A . 5. Suppose that A ∈ R n and the characteristic polynomial of A is ( t 2 + 4) 2 ( t1) 2 . Is A symmetric? Why or why not. 6. Let T be a linear transformation on C n . Recall that the adjoint of T , denoted T * satisﬁes the equation h Tx,y i = h x,T * y i for all x,y ∈ C n . (a) If T = T * , show that the eigenvalues of T are real. (b) If TT * = T * T , then ker( T k ) = ker( T ) for all natural numbers k . (c) If T * T = TT * , then ker( T ) = ker( T * ). (d) In general show that λ is an eigenvalue of T if and only if λ is an eigenvalue of T * . 1...
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 Fall '08
 STAPLES
 Linear Algebra, Algebra, Eigenvectors, Vectors, Characteristic polynomial, Orthogonal matrix, a. compute

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