Chapter 9

Let \(T: V \rightarrow V\) be a linear operator. Given \(\mathbf{v}\) in \(V,\) let \(U\) denote the set of vectors in \(V\) that lie in every \(T\) -invariant subspace that contains \(\mathbf{v}\). a. Show that \(U\) is a \(T\) -invariant subspace of \(V\) containing \(\mathbf{v}\). b. Show that \(U\) is contained in every \(T\) -invariant subspace of \(V\) that contains \(\mathbf{v}\).

Exercise 9.3 .6 Show that the only subspaces of \(V\) that are \(T\) -invariant for every operator \(T: V \rightarrow V\) are 0 and \(V\). Assume that \(V\) is finite dimensional.

Find \(P_{D \leftarrow B}\) if \(B=\left\\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}, \mathbf{b}_{4}\right\\}\) and \(D=\left\\{\mathbf{b}_{2}, \mathbf{b}_{3}, \mathbf{b}_{1}, \mathbf{b}_{4}\right\\} .\) Change matrices arising when the bases differ only in the order of the vectors are called permutation matrices.

Suppose that \(T: V \rightarrow V\) is a linear operator and that \(U\) is a \(T\) -invariant subspace of \(V .\) If \(S\) is an invertible operator, put \(T^{\prime}=S T S^{-1}\). Show that \(S(U)\) is a \(T^{\prime}\) -invariant subspace.

In each case, find \(P=P_{B_{0} \leftarrow B}\) and verify that \(P^{-1} M_{B_{0}}(T) P=M_{B}(T)\) for the given operator \(T\). a. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}, T(a, b, c)=(2 a-b, b+c, c-3 a) ;\) \(B_{0}=\\{(1,1,0),(1,0,1),(0,1,0)\\}\) and \(B\) is the standard basis. b. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2},\) \(\quad T\left(a+b x+c x^{2}\right)=(a+b)+(b+c) x+(c+a) x^{2}\) \(B_{0}=\left\\{1, x, x^{2}\right\\}\) and \(B=\left\\{1-x^{2}, 1+x, 2 x+x^{2}\right\\}\) c. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) \(T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left[\begin{array}{ll}a+d & b+c \\ a+c & b+d\end{array}\right]\) \(B_{0}=\left\\{\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right\\}\) and \(B=\left\\{\left[\begin{array}{ll}1 & 1 \\\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right]\right\\}\)

In each case, find \(T^{-1}\) and verify that \(\left[M_{D B}(T)\right]^{-1}=M_{B D}\left(T^{-1}\right)\). a. \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, T(a, b)=(a+2 b, 2 a+5 b) ;\) \(\quad B=D=\) standard b. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}, T(a, b, c)=(b+c, a+c, a+b) ;\) \(B=D=\) standard c. \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{3}, T\left(a+b x+c x^{2}\right)=(a-c, b, 2 a-c) ;\) \(\quad B=\left\\{1, x, x^{2}\right\\}, D=\) standard d. \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{3},\) \(\quad T\left(a+b x+c x^{2}\right)=(a+b+c, b+c, c)\) \(\quad B=\left\\{1, x, x^{2}\right\\}, D=\) standard

In each case, show that \(U\) is \(T\) -invariant, use it to find a block upper triangular matrix for \(T,\) and use that to compute \(c_{T}(x)\). a. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\) $$\begin{aligned} T(&\left.a+b x+c x^{2}\right) \\ &=(-a+2 b+c)+(a+3 b+c) x+(a+4 b) x^{2} \\ U &=\operatorname{span}\left\\{1, x+x^{2}\right\\} \end{aligned}$$ b. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\) $$\begin{aligned} T &\left(a+b x+c x^{2}\right) \\ &=(5 a-2 b+c)+(5 a-b+c) x+(a+2 c) x^{2} \\ U&=\operatorname{span}\left\\{1-2 x^{2}, x+x^{2}\right\\} \end{aligned}$$

In each case, show that \(M_{D B}(T)\) is invertible and use the fact that \(M_{B D}\left(T^{-1}\right)=\left[M_{B D}(T)\right]^{-1}\) to determine the action of \(T^{-1}\). a. \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{3}, T\left(a+b x+c x^{2}\right)=(a+c, c, b-c) ;\) \(B=\left\\{1, x, x^{2}\right\\}, D=\) standard b. \(T: \mathbf{M}_{22} \rightarrow \mathbb{R}^{4}\) \(T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=(a+b+c, b+c, c, d)\) \(B=\left\\{\begin{array}{l}\left.\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right\\} \\ D=\text { standard }\end{array}\right.\)

In each case, verify that \(P^{-1} A P=D\) and find a basis \(B\) of \(\mathbb{R}^{2}\) such that \(M_{B}\left(T_{A}\right)=D\). $$\begin{array}{l}\text { a. } A=\left[\begin{array}{ll}11 & -6 \\\12 & -6\end{array}\right] P=\left[\begin{array}{ll}2 & 3 \\\3 & 4\end{array}\right] D=\left[\begin{array}{ll}2 & 0 \\\0 & 3\end{array}\right] \\\\\text { b.}A=\left[\begin{array}{ll}29 & -12 \\\70 & -29\end{array}\right] P=\left[\begin{array}{ll}3 & 2 \\\7 & 5 \end{array}\right] D=\left[\begin{array}{ll}1 & 0 \\\0 & -1\end{array}\right]\end{array}$$

In each case, compute the characteristic polynomial \(c_{T}(x)\). a. \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, T(a, b)=(a-b, 2 b-a)\) b. \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}, T(a, b)=(3 a+5 b, 2 a+3 b)\) c. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\) $$\begin{aligned} T(&\left.a+b x+c x^{2}\right) \\ &=(a-2 c)+(2 a+b+c) x+(c-a) x^{2} \end{aligned} $$ d. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\) $$\begin{aligned} T &\left(a+b x+c x^{2}\right) \\ &=(a+b-2 c)+(a-2 b+c) x+(b-2 a) x^{2} \end{aligned}$$ e. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}, T(a, b, c)=(b, c, a)\) f. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}, T\left[\begin{array}{ll}a & b \\\ c & d\end{array}\right]=\left[\begin{array}{ll}a-c & b-d \\ a-c & b-d\end{array}\right]\)