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HW7solutions_2

# HW7solutions_2 - MATH 115 SOLUTION SET 7 ANSWERS TO...

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MATH 115 SOLUTION SET 7 ANSWERS TO SELECTED PROBLEMS § 5.1 7. The point of this problem is to extend the definition of determinant from matrices to linear transformations. The result is always that a property satisfied by matrices is always satisfied by linear transformations as well: a. If Q is the change of basis matrix from β to γ , then [ T ] β = Q [ T ] γ Q - 1 , so that [ T ] β and [ T ] γ are similar and therefore have the same determinant. b. T is invertible if and only if the matrix [ T ] β is invertible, which is so if and only if det[ T ] β 6 = 0. But det[ T ] β is the same as det T . c. We have det( T - 1 ) = det[ T - 1 ] β = det[ T ] - 1 β = (det T ) - 1 . d. We have det( TU ) = det([ TU ] β ) = det([ T ] β [ U ] β ) = det([ T ] β ) det([ U ] β ) = det( TU ). e. We have det( T - λI V ) = det([ T - λI V ] β ) = det([ T ] - λI ). The second equality comes from Thm. 2.8. / 8. a. T is invertible if and only if det T 6 = 0, which is true if and only if 0 is not a root of the characteristic polynomial det( T - λI V ), which is true if and only if 0 is not an eigenvalue of T . b. We have that λ is an eigenvalue of T if and only if for some v V , we have Tv = λv . But since T is invertible, every eigenvalue is nonzero and we may rewrite
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