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**Unformatted text preview: **MAT 442: Test 3 Solutions 1. For each part, state the definition or theorem for linear operators. (a) Define eigenvalue and eigenvector If T is an operator on a vector space V over F , an eigenvector x V is a non-zero vector such that T ( x ) = x for some F . A scalar is an eigenvalue if there is a corresponding eigenvector x as above. (b) Define eigenspace If F is an eigenvalue of a linear operator T on a vector space V , the - eigenspace is the set { v V | T ( v ) = v } . (c) Define characteristic polynomial, including an explanation as to why the definition is well-defined. The characteristic polynomial of a linear operator T on a finite dimensional vector space V is det([ T ] - tI ) where is a basis for V . If we chose a different basis , then there would exist an invertible matrix Q such that Q- 1 [ T ] Q = [ T ] , which implies Q- 1 ([ T ] - tI ) Q = [ T ] - tI , and in general, det( Q- 1 BQ ) = det( B ), so the characteristic polynomial is independent of the choice of basis. (d) State the Cayley-Hamilton theorem If T is a linear operator on a finite dimensional vector space V and T ( t ) is its characteristic polynomial, then...

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