# Tag Archives: Exercise D

## Chapter 8 Exercise D

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## Chapter 7 Exercise D

11. Solution: It follows from 7.45 that $T=S\sqrt{T^*T}$ for an isometry $S\in\ca L(V)$. Note that $\sqrt{T^*T}$ is self-adjoint and $S^{-1}=S^*$ (7.42 (e), (f)), we have \[ TT^*=S\sqrt{T^*T}(S\sqrt{T^*T})^*=S\sqrt{T^*T}\sqrt{T^*T}S^*=S(T^*T)S^{-1}. \]It follows from Problem 15 of Exercise 5A that $TT^*$ and $T^*T$ have … Continue reading

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## Chapter 3 Exercise D

1. Suppose $T\in\ca L(U, V)$ and $S\in\ca L(V, W)$ are both invertible linear maps. Prove that $ST\in\ca L(U, W)$ / is invertible and that $(ST)^{-1}=T^{-1}S^{-1}$. Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 22. It is almost … Continue reading

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