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(
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This note was uploaded on 02/01/2010 for the course MATH math115a taught by Professor Shalom during the Spring '10 term at UCLA.