91Ó°ÊÓ

Label the following statements as true or false. (a) Every linear operator on an \(n\)-dimensional vector space has \(n\) distinct eigenvalues. (b) If a real matrix has one eigenvector, then it has an infinite number of eigenvectors. (c) There exists a square matrix with no eigenvectors. (d) Eigenvalues must be nonzero scalars. (e) Any two eigenvectors are linearly independent. (f) The sum of two eigenvalues of a linear operator \(T\) is also an eigenvalue of \(T\). (g) Linear operators on infinite-dimensional vector spaces never have eigenvalues. (h) An \(n \times n\) matrix \(A\) with entries from a field \(F\) is similar to a diagonal matrix if and only if there is a basis for \(\mathrm{F}^{n}\) consisting of eigenvectors of \(A\). (i) Similar matrices always have the same eigenvalues. (j) Similar matrices always have the same eigenvectors. (k) The sum of two eigenvectors of an operator \(\mathrm{T}\) is always an eigenvector of \(T\).

For each of the following linear operators \(\mathrm{T}\) on a vector space \(\mathrm{V}\), compute the determinant of \(\mathrm{T}\) and the characteristic polynomial of \(\mathrm{T}\). (a) \(\mathrm{V}=\mathrm{R}^{2}, \mathrm{~T}\left(\begin{array}{l}a \\\ b\end{array}\right)=\left(\begin{array}{c}2 a-b \\ 5 a+3 b\end{array}\right)\) (b) \(\mathbf{V}=\mathbf{R}^{3}, \mathbf{T}\left(\begin{array}{l}a \\ b \\\ c\end{array}\right)=\left(\begin{array}{c}a-3 b+2 c \\ -2 a+b+c \\ 4 a-c\end{array}\right)\)Sec. \(5.1\) Eigenvalues and Eigenvectors 257 (c) \(\mathrm{V}=\mathrm{P}_{3}(R)\), \(\mathrm{T}\left(a+b x+c x^{2}+d x^{3}\right)=(a-c)+(-a+b+d) x+(a+b-d) x^{2}-c x^{3}\) (d) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(R), \mathrm{T}(A)=2 A^{t}-A\)

Let \(\mathrm{T}\) be a linear operator on a finite-dimensional vector space \(\mathrm{V}\). Prove that the following subspaces are \(T\)-invariant. (a) \(\\{0\\}\) and \(\mathrm{V}\) (b) \(N(T)\) and \(R(T)\) (c) \(E_{\lambda}\), for any eigenvalue \(\lambda\) of \(T\)

For each of the following linear operators \(T\) on a vector space \(V\) and ordered bases \(\beta\), compute \([T]_{\beta}\), and determine whether \(\beta\) is a basis consisting of eigenvectors of \(T\). (a) \(\mathrm{V}=\mathrm{R}^{2}, \mathrm{~T}\left(\begin{array}{l}a \\\ b\end{array}\right)=\left(\begin{array}{l}10 a-6 b \\ 17 a-10 b\end{array}\right)\), and \(\beta=\left\\{\left(\begin{array}{l}1 \\\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 3\end{array}\right)\right\\}\) (b) \(\quad \mathrm{V}=\mathrm{P}_{1}(R), \mathrm{T}(a+b x)=(6 a-6 b)+(12 a-11 b) x\), and $$ \beta=\\{3+4 x, 2+3 x\\} $$ (c) \(\mathrm{V}=\mathrm{R}^{3}, \mathrm{~T}\left(\begin{array}{l}a \\ b \\\ c\end{array}\right)=\left(\begin{array}{r}3 a+2 b-2 c \\ -4 a-3 b+2 c \\\ -c\end{array}\right)\), and $$ \beta=\left\\{\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 2 \end{array}\right)\right\\} $$ (d) \(\mathrm{V}=\mathrm{P}_{2}(R), \mathrm{T}\left(a+b x+c x^{2}\right)=\) $$ (-4 a+2 b-2 c)-(7 a+3 b+7 c) x+(7 a+b+5 c) x^{2}, $$ and \(\beta=\left\\{x-x^{2},-1+x^{2},-1-x+x^{2}\right\\}\) (e) \(\mathrm{V}=\mathrm{P}_{3}(R), \mathrm{T}\left(a+b x+c x^{2}+d x^{3}\right)=\) $$ -d+(-c+d) x+(a+b-2 c) x^{2}+(-b+c-2 d) x^{3}, $$ and \(\beta=\left\\{1-x+x^{3}, 1+x^{2}, 1, x+x^{2}\right\\}\) (f) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(R), \mathrm{T}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}-7 a-4 b+4 c-4 d & b \\\ -8 a-4 b+5 c-4 d & d\end{array}\right)\), and $$ \beta=\left\\{\left(\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right),\left(\begin{array}{rr} -1 & 2 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 1 & 0 \\ 2 & 0 \end{array}\right),\left(\begin{array}{rr} -1 & 0 \\ 0 & 2 \end{array}\right)\right\\} $$

Prove the matrix version of the corollary to Theorem 5.5: If \(A \in\) \(\mathrm{M}_{n \times n}(F)\) has \(n\) distinct eigenvalues, then \(A\) is diagonalizable.

For each of the following matrices \(A \in \mathrm{M}_{n \times n}(F)\), (i) Determine all the eigenvalues of \(A\). (ii) For each eigenvalue \(\lambda\) of \(A\), find the set of eigenvectors corresponding to \(\lambda\). (iii) If possible, find a basis for \(\mathrm{F}^{n}\) consisting of eigenvectors of \(A\). (iv) If successful in finding such a basis, determine an invertible matrix \(Q\) and a diagonal matrix \(D\) such that \(Q^{-1} A Q=D\).258 Chap. 5 Diagonalization (a) \(A=\left(\begin{array}{ll}1 & 2 \\ 3 & 2\end{array}\right) \quad\) for \(F=R\) (b) \(A=\left(\begin{array}{rrr}0 & -2 & -3 \\ -1 & 1 & -1 \\ 2 & 2 & 5\end{array}\right) \quad\) for \(F=R\) (c) \(A=\left(\begin{array}{rr}i & 1 \\ 2 & -i\end{array}\right) \quad\) for \(F=C\) (d) \(A=\left(\begin{array}{ccc}2 & 0 & -1 \\ 4 & 1 & -4 \\ 2 & 0 & -1\end{array}\right) \quad\) for \(F=R\)

Prove that if \(A \in \mathbf{M}_{n \times n}(C)\) is diagonalizable and \(L=\lim _{m \rightarrow \infty} A^{m}\) exists, then either \(L=I_{n}\) or \(\operatorname{rank}(L)

Let \(\mathrm{T}\) be a linear operator on a vector space \(\mathrm{V}\). Prove that the intersection of any collection of \(T\)-invariant subspaces of \(V\) is a \(T\)-invariant subspace of V.

For each linear operator \(T\) on \(V\), find the eigenvalues of \(T\) and an ordered basis \(\beta\) for \(\mathrm{V}\) such that \([\mathrm{T}]_{\beta}\) is a diagonal matrix. (a) \(\mathrm{V}=\mathrm{R}^{2}\) and \(\mathrm{T}(a, b)=(-2 a+3 b,-10 a+9 b)\) (b) \(\mathrm{V}=\mathrm{R}^{3}\) and \(\mathrm{T}(a, b, c)=(7 a-4 b+10 c, 4 a-3 b+8 c,-2 a+b-2 c)\) (c) \(\mathrm{V}=\mathrm{R}^{3}\) and \(\mathrm{T}(a, b, c)=(-4 a+3 b-6 c, 6 a-7 b+12 c, 6 a-6 b+11 c)\) (d) \(\mathrm{V}=\mathrm{P}_{1}(R)\) and \(\mathrm{T}(a x+b)=(-6 a+2 b) x+(-6 a+b)\) (e) \(\mathrm{V}=\mathrm{P}_{2}(R)\) and \(\mathrm{T}(f(x))=x f^{\prime}(x)+f(2) x+f(3)\) (f) \(\mathrm{V}=\mathrm{P}_{3}(R)\) and \(\mathrm{T}(f(x))=f(x)+f(2) x\) (g) \(\mathbf{V}=\mathbf{P}_{3}(R)\) and \(\mathbf{T}(f(x))=x f^{\prime}(x)+f^{\prime \prime}(x)-f(2)\) (h) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(R)\) and \(\mathrm{T}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}d & b \\ c & a\end{array}\right)\) (i) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(R)\) and \(\mathrm{T}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}c & d \\ a & b\end{array}\right)\) (j) \(\mathrm{V}=\mathrm{M}_{2 \times 2}(R)\) and \(\mathrm{T}(A)=A^{t}+2 \cdot \operatorname{tr}(A) \cdot I_{2}\)

Prove that the restriction of a linear operator \(\mathrm{T}\) to a \(\mathrm{T}\)-invariant subspace is a linear operator on that subspace.