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Problem 6. Let V be a finite dimensional vector space over F. This problem explores some consequences of the fact that T = L(V) is invertible iff it is injective iff it is surjective. (a) Let T, SE L(V). Prove that ST is invertible if and only if both S and T are invertible. (b) Let T, SE L(V) be such that ST I. Prove that TS I as well, and therefore, S and T are inverses of each other. (c) Let S, T, UEL(V) be such that STU - I. Prove that T is invertible and T-1 = US.

Problem 6. Let V be a finite dimensional vector space over F. This problem explores some consequences of the fact that T = L(V) is invertible iff it is injective iff it is surjective. (a) Let T, SE L(V). Prove that ST is invertible if and only if both S and T are invertible. (b) Let T, SE L(V) be such that ST I. Prove that TS I as well, and therefore, S and T are inverses of each other. (c) Let S, T, UEL(V) be such that STU - I. Prove that T is invertible and T-1 = US.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 6.** Let \( V \) be a finite dimensional vector space over \( \mathbb{F} \). This problem explores some consequences of the fact that \( T \in \mathcal{L}(V) \) is invertible iff it is injective iff it is surjective. (a) Let \( T, S \in \mathcal{L}(V) \). Prove that \( ST \) is invertible if and only if both \( S \) and \( T \) are invertible. (b) Let \( T, S \in \mathcal{L}(V) \) be such that \( ST = I \). Prove that \( TS = I \) as well, and therefore, \( S \) and \( T \) are inverses of each other. (c) Let \( S, T, U \in \mathcal{L}(V) \) be such that \( STU = I \). Prove that \( T \) is invertible and \( T^{-1} = US \).
Transcribed Image Text:**Problem 6.** Let \( V \) be a finite dimensional vector space over \( \mathbb{F} \). This problem explores some consequences of the fact that \( T \in \mathcal{L}(V) \) is invertible iff it is injective iff it is surjective. (a) Let \( T, S \in \mathcal{L}(V) \). Prove that \( ST \) is invertible if and only if both \( S \) and \( T \) are invertible. (b) Let \( T, S \in \mathcal{L}(V) \) be such that \( ST = I \). Prove that \( TS = I \) as well, and therefore, \( S \) and \( T \) are inverses of each other. (c) Let \( S, T, U \in \mathcal{L}(V) \) be such that \( STU = I \). Prove that \( T \) is invertible and \( T^{-1} = US \).
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