91Ó°ÊÓ

Prove that for any matrix \(A \in \mathrm{M}_{m \times n}(F),\left(\mathrm{R}\left(\mathrm{L}_{A^{*}}\right)\right)^{\perp}=\mathrm{N}\left(\mathrm{L}_{A}\right)\).

Let \(\left\\{v_{1}, v_{2}, \ldots, v_{k}\right\\}\) be an orthogonal set in \(\mathrm{V}\), and let \(a_{1}, a_{2}, \ldots, a_{k}\) be scalars. Prove that $$ \left\|\sum_{i=1}^{k} a_{i} v_{i}\right\|^{2}=\sum_{i=1}^{k}\left|a_{i}\right|^{2}\left\|v_{i}\right\|^{2} . $$

Let \(V\) be an inner product space, and let \(T\) be a linear operator on \(V\). Prove the following results. (a) \(R\left(T^{*}\right)^{\perp}=N(T)\). (b) If \(V\) is finite-dimensional, then \(R\left(T^{*}\right)=N(T)^{\perp}\).

Prove that the relation of congruence is an equivalence relation.

Let \(\mathrm{T}\) be a normal operator on a finite-dimensional real inner product space \(V\) whose characteristic polynomial splits. Prove that \(V\) has an orthonormal basis of eigenvectors of \(\mathrm{T}\). Hence prove that \(\mathrm{T}\) is selfadjoint.

An \(n \times n\) real matrix \(A\) is said to be a Gramian matrix if there exists a real (square) matrix \(B\) such that \(A=B^{t} B\). Prove that \(A\) is a Gramian matrix if and only if \(A\) is symmetric and all of its eigenvalues are nonnegative. Hint: Apply Theorem \(6.17\) to \(\mathrm{T}=\mathrm{L}_{A}\) to obtain an orthonormal basis \(\left\\{v_{1}, v_{2}, \ldots, v_{n}\right\\}\) of eigenvectors with the associated eigenvalues \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n} .\) Define the linear operator \(\mathrm{U}\) by \(\mathrm{U}\left(v_{i}\right)=\sqrt{\lambda_{i}} v_{i}\).

Suppose that \(\langle\cdot, \cdot\rangle_{1}\) and \(\langle\cdot, \cdot\rangle_{2}\) are two inner products on a vector space V. Prove that \(\langle\cdot, \cdot\rangle=\langle\cdot, \cdot\rangle_{1}+\langle\cdot, \cdot\rangle_{2}\) is another inner product on \(\mathrm{V}\).

Let \(\mathrm{V}\) be an inner product space, \(S\) and \(S_{0}\) be subsets of \(\mathrm{V}\), and W be a finite-dimensional subspace of V. Prove the following results. (a) \(S_{0} \subseteq S\) implies that \(S^{\perp} \subseteq S_{0}^{\perp}\). (b) \(S \subseteq\left(S^{\perp}\right)^{\perp}\); so \(\operatorname{span}(S) \subseteq\left(S^{\perp}\right)^{\perp}\). (c) \(\mathrm{W}=\left(\mathrm{W}^{\perp}\right)^{\perp}\). Hint: Use Exercise 6 . (d) \(\vee=W \oplus W^{\perp}\). (See the exercises of Section 1.3.)

Prove that if \(A\) and \(B\) are unitarily equivalent matrices, then \(A\) is positive definite [semidefinite] if and only if \(B\) is positive definite [semidefinite]. (See the definitions in the exercises in Section 6.4.)

Let \(W_{1}\) and \(W_{2}\) be subspaces of a finite-dimensional inner product space. Prove that \(\left(\mathrm{W}_{1}+\mathrm{W}_{2}\right)^{\perp}=\mathrm{W}_{1}^{\perp} \cap \mathrm{W}_{2}^{\perp}\) and \(\left(\mathrm{W}_{1} \cap \mathrm{W}_{2}\right)^{\perp}=\mathrm{W}_{1}^{\perp}+\mathrm{W}_{2}^{\perp} .\) (See the definition of the sum of subsets of a vector space on page 22.) Hint for the second equation: Apply Exercise 13(c) to the first equation.