91Ó°ÊÓ

Let \(\beta=\left\\{v_{1}, v_{2}, \ldots, v_{n}\right\\}\) be a basis for a vector space \(\mathrm{V}\) and \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) be a linear transformation. Prove that \([\mathrm{T}]_{\beta}\) is upper triangular if and only if \(\mathrm{T}\left(v_{j}\right) \in \operatorname{span}\left(\left\\{v_{1}, v_{2}, \ldots, v_{j}\right\\}\right)\) for \(j=1,2, \ldots, n\). Visit goo.gl/k9ZrQb for a solution.

Let \(\mathrm{V}\) be a finite-dimensional vector space over a field \(F\), and let \(\beta=\) \(\left\\{x_{1}, x_{2}, \ldots, x_{n}\right\\}\) be an ordered basis for \(\mathrm{V}\). Let \(Q\) be an \(n \times n\) invertible matrix with entries from \(F\). Define $$ x_{j}^{\prime}=\sum_{i=1}^{n} Q_{i j} x_{i} \quad \text { for } 1 \leq j \leq n, $$ and set \(\beta^{\prime}=\left\\{x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n}^{\prime}\right\\} .\) Prove that \(\beta^{\prime}\) is a basis for \(\mathrm{V}\) and hence that \(Q\) is the change of coordinate matrix changing \(\beta^{\prime}\)-coordinates into \(\beta\)-coordinates. Visit goo.gl/vsxsGH for a solution.

Denotes a finite-dimensional vector space over \(F\). For every subset \(S\) of \(\mathrm{V}\), define the annihilator \(S^{0}\) of \(S\) as $$ S^{0}=\left\\{\mathrm{f} \in \mathrm{V}^{*}: \mathrm{f}(x)=0 \text { for all } x \in S\right\\} . $$ (a) Prove that \(S^{0}\) is a subspace of \(\mathrm{V}^{*}\). (b) If \(\mathrm{W}\) is a subspace of \(\mathrm{V}\) and \(x \notin \mathrm{W}\), prove that there exists \(\mathrm{f} \in \mathrm{W}^{0}\) such that \(\mathrm{f}(x) \neq 0\). (c) Prove that \(\left(S^{0}\right)^{0}=\operatorname{span}(\psi(S))\), where \(\psi\) is defined as in Theorem \(2.26\). (d) For subspaces \(W_{1}\) and \(W_{2}\), prove that \(W_{1}=W_{2}\) if and only if \(\mathrm{W}_{1}^{0}=\mathrm{W}_{2}^{0}\). (e) For subspaces \(\mathrm{W}_{1}\) and \(\mathrm{W}_{2}\), show that \(\left(\mathrm{W}_{1}+\mathrm{W}_{2}\right)^{0}=\mathrm{W}_{1}^{0} \cap \mathrm{W}_{2}^{0}\).

Prove that if \(W\) is a subspace of \(V\), then \(\operatorname{dim}(W)+\operatorname{dim}\left(W^{0}\right)=\operatorname{dim}(V)\). Hint: Extend an ordered basis \(\left\\{x_{1}, x_{2}, \ldots, x_{k}\right\\}\) of \(\mathrm{W}\) to an ordered basis \(\beta=\left\\{x_{1}, x_{2}, \ldots, x_{n}\right\\}\) of \(\mathrm{V}\). Let \(\beta^{*}=\left\\{\mathrm{f}_{1}, \mathrm{f}_{2}, \ldots, \mathrm{f}_{n}\right\\}\). Prove that \(\left\\{\mathrm{f}_{k+1}, \mathrm{f}_{k+2}, \ldots, \mathrm{f}_{n}\right\\}\) is a basis for \(\mathrm{W}^{0}\).

Let \(\mathrm{V}\) and \(\mathrm{W}\) be \(n\)-dimensional vector spaces, and let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) be a linear transformation. Suppose that \(\beta\) is a basis for \(\mathrm{V}\). Prove that \(\mathrm{T}\) is an isomorphism if and only if \(\mathrm{T}(\beta)\) is a basis for \(\mathrm{W}\).

Recall the definition of \(\mathrm{P}(R)\) on page 11. Define $$ \mathrm{T}: \mathrm{P}(R) \rightarrow \mathrm{P}(R) \quad \text { by } \quad \mathrm{T}(f(x))=\int_{0}^{x} f(t) d t . $$ Prove that \(\mathrm{T}\) linear and one-to-one, but not onto.

Let \(A\) and \(B\) be matrices for which the product matrix \(A B\) is defined, and let \(u_{j}\) and \(v_{j}\) denote the \(j\) th columns of \(A B\) and \(B\), respectively. If \(v_{p}=c_{1} v_{j_{1}}+c_{2} v_{j_{2}}+\cdots+c_{k} v_{j_{k}}\) for some scalars \(c_{1}, c_{2}, \ldots c_{k}\), prove that \(u_{p}=c_{1} u_{j_{1}}+c_{2} u_{j_{2}}+\cdots+c_{k} u_{j_{k}} .\) Visit goo.gl/sRpves for a solution.

Let \(\mathrm{T}: \mathrm{P}(R) \rightarrow \mathrm{P}(R)\) be defined by \(\mathrm{T}(f(x))=f^{\prime}(x)\). Recall that \(\mathrm{T}\) is linear. Prove that \(\mathrm{T}\) is onto, but not one-to-one.

\(^{\dagger}\) Let \(\mathrm{V}\) and \(\mathrm{W}\) be finite-dimensional vector spaces and \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) be an isomorphism. Let \(V_{0}\) be a subspace of \(V\). (a) Prove that \(T\left(V_{0}\right)\) is a subspace of \(W\). (b) Prove that \(\operatorname{dim}\left(\mathrm{V}_{0}\right)=\operatorname{dim}\left(\mathrm{T}\left(\mathrm{V}_{0}\right)\right)\).

For the definition of projection and related facts, see pages 76-77. Let \(\mathrm{V}\) be a vector space and \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) be a linear transformation. Prove that \(\mathrm{T}=\mathrm{T}^{2}\) if and only if \(\mathrm{T}\) is a projection on \(\mathrm{W}_{1}=\\{y: \mathrm{T}(y)=y\\}\) along \(N(T)\).