MA353_HW8_sol

# MA353_HW8_sol - Purdue University MA 353 Linear Algebra II...

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Purdue University MA 353: Linear Algebra II with Applications Homework 8, due Mar. 12, Some Solutions (I) Sec. 5.1#8 (a) Prove that a linear operator T on a ﬁnite-dimensional vector space is invertible if and only if zero is not an eigenvalue of T . Proof. Let us call this ﬁnite-dimensional vector space V . (= ) Assume that T is invertible, i.e. T - 1 exists. Let λ be an eigenvalue of T . Then T ( v ) = λv for some non-zero v V . By applying T - 1 to both sides, we get T - 1 T ( v ) = T - 1 ( λv ), which gives v = λT - 1 ( v ). Now if λ = 0, then this would give v = 0, but v is not zero, which is a contradiction. Hence λ 6 = 0. ( =) Assume zero is not an eigenvalue of T . Then N ( T ) = { 0 } because if N ( T ) 6 = { 0 } , then there is non-zero v V such that T ( v ) = 0, i.e. T ( v ) = 0 F v , which would imply that 0 F is an eigenvalue. Hence T is 1-1. But V is ﬁnite dimensional, this implies that T is invertible. ± (b) Let

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MA353_HW8_sol - Purdue University MA 353 Linear Algebra II...

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