# 7.E: Exercises for Chapter 7

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## Calculational Exercises

1. Let $$T \in \cal{L}(\mathbb{F}^2 , \mathbb{F}^2)$$ be deﬁned by

$T (u, v) = (v, u)$

for every $$u, v \in \mathbb{F}.$$ Compute the eigenvalues and associated eigenvectors for $$T.$$

2. Let $$T \in \cal{L}(\mathbb{F}^3 , \mathbb{F}^3$$) be deﬁned by

$T (u, v, w) = (2v, 0, 5w)$

for every $$u, v, w \in \mathbb{F}.$$ Compute the eigenvalues and associated eigenvectors for $$T.$$

3. Let $$n \in \mathbb{Z}_+$$ be a positive integer and $$T \in \cal{L}(\mathbb{F}^n , \mathbb{F}^n )$$ be deﬁned by

$T (x_1 ,\ldots, x_n ) = (x1 + \cdots + x_n , \ldots, x_1 + \cdots + x_n)$

for every $$x_1 ,\ldots, x_n \in \mathbb{F}.$$ Compute the eigenvalues and associated eigenvectors for $$T.$$

4. Find eigenvalues and associated eigenvectors for the linear operators on $$\mathbb{F}^2$$ deﬁned by each given $$2 \times 2$$ matrix.

$$(a) \left[ \begin{array}{cc} 3 & 0 \\ 8 & -1 \end{array} \right], ~~ (b) \left[ \begin{array}{cc} 10 & -9 \\ 4 & -2 \end{array} \right], ~~ (c) \left[ \begin{array}{cc} 0 & 3 \\ 4 & 0 \end{array} \right],$$

$$(d) \left[ \begin{array}{cc} -2 & -7 \\ 1 & 2 \end{array} \right], ~~ (e) \left[ \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right], ~~ (f) \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]$$

Hint: Use the fact that, given a matrix $$A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in \mathbb{F}^{2\times2} , \lambda \in \mathbb{F}$$ is an eigenvalue for A if and only if $$(a − \lambda)(d − \lambda) − bc = 0.$$

5. For each matrix $$A$$ below, ﬁnd eigenvalues for the induced linear operator $$T$$ on $$\mathbb{F}^n$$ without performing any calculations. Then describe the eigenvectors $$v \in \mathbb{F}^n$$ associated to each eigenvalue $$\lambda$$ by looking at solutions to the matrix equation $$(A − \lambda I)v = 0,$$ where I denotes the identity map on $$\mathbb{F}^n.$$

$$(a) \left[ \begin{array}{cc} -1 & 6 \\ 0 & 5 \end{array} \right], ~~(b) \left[ \begin{array}{cccc} -\frac{1}{3} & 0 & 0 & 0 \\ 0 & -\frac{1}{3} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{1}{2} \end{array} \right], ~~(c) \left[ \begin{array}{cccc} 1 & 3 & 7 & 11 \\ 0 & \frac{1}{2} & 3 & 8 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 2 \end{array} \right]$$

6. For each matrix $$A$$ below, describe the invariant subspaces for the induced linear operator $$T on \mathbb{F}^2$$ that maps each $$v \in \mathbb{F}^2$$ to $$T (v) = Av.$$

$$(a) \left[ \begin{array}{cc} 4 & -1 \\ 2 & 1 \end{array} \right], ~~ (b) \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right], ~~ (c) \left[ \begin{array}{cc} 2 & 3 \\ 0 & 2 \end{array} \right], ~~ (d) \left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right]$$

7. Let $$T \in \cal{L}(\mathbb{R}^2)$$ be deﬁned by

$T \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} x \\ x+y \end{array} \right),~~\rm{~for ~all~} \left( \begin{array}{c} x \\ y \end{array} \right) \in \mathbb{R}^2.$

Deﬁne two real numbers $$\lambda_+$$ and $$\lambda_−$$ as follows:

$\lambda_+=\frac{1+\sqrt{5}}{2}, ~~ \lambda_-=\frac{1-\sqrt{5}}{2}.$

(a) Find the matrix of $$T$$ with respect to the canonical basis for $$\mathbb{R}^2$$ (both as the domain and the codomain of $$T$$ ; call this matrix $$A$$).
(b) Verify that $$\lambda_+$$ and $$\lambda_−$$ are eigenvalues of $$T$$ by showing that $$v_+$$ and $$v_−$$ are eigen-
vectors, where

$v_+ = \left( \begin{array}{c} 1 \\ \lambda_+ \end{array} \right),~~ v_- = \left( \begin{array}{c} 1 \\ \lambda_- \end{array} \right).$

(c) Show that $$(v_+ , v_− )$$ is a basis of $$\mathbb{R}^2.$$
(d) Find the matrix of $$T$$ with respect to the basis $$(v_+ , v_− )$$ for $$\mathbb{R}^2$$ (both as the domain
and the codomain of $$T$$ ; call this matrix $$B$$).

## Proof-Writing Exercises

1. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F}$$ with $$T \in \cal{L}$$$$(V, V),$$ and let $$U_1 , \ldots, U_m$$ be subspaces of $$V$$ that are invariant under $$T$$. Prove that $$U_1 + \cdots + U_m$$ must then also be an invariant subspace of $$V$$ under $$T .$$

2. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F }$$ with $$T \in \cal{L}$$$$(V, V ),$$ and suppose that $$U_1$$ and $$U_2$$ are subspaces of $$V$$ that are invariant under $$T$$. Prove that $$U_1 \cap U_2$$ is also an invariant subspace of $$V$$ under $$T.$$

3. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F}$$ with $$T \in \cal{L}$$$$(V, V)$$ invertible and $$\lambda \in \mathbb{F} = \{0\}.$$ Prove $$\lambda$$ is an eigenvalue for $$T$$ if and only if $$\lambda−1$$ is an eigenvalue for $$T −1.$$

4. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F},$$ and suppose that $$T \in \cal{L}$$$$(V, V)$$ has the property that every $$v \in V$$ is an eigenvector for $$T$$ . Prove that $$T$$ must then be a scalar multiple of the identity function on $$V.$$

5. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F},$$ and let $$S, T \in \cal{L}$$$$(V)$$ be linear operators on $$V$$ with $$S$$ invertible. Given any polynomial $$p(z) \in \mathbb{F}[z],$$ prove that

$p(S \circ T \circ S^{ −1} ) = S \circ p(T ) \circ S^{ −1}.$

6. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{C}, T \in \cal{L}$$$$(V)$$ be a linear operator on $$V$$ , and $$p(z) \in \mathbb{C}[z]$$ be a polynomial. Prove that $$\lambda \in \mathbb{C}$$ is an eigenvalue of the linear operator $$p(T ) \in \cal{L}$$$$(V)$$ if and only if $$T$$ has an eigenvalue $$\mu \in \mathbb{C}$$ such that $$p(\mu) = \lambda.$$

7. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{C}$$ with $$T \in \cal{L}$$$$(V)$$ a linear operator on $$V.$$ Prove that, for each $$k = 1,\ldots , dim(V ),$$ there is an invariant subspace $$U_k$$ of $$V$$ under $$T$$ such that $$dim(U_k ) = k.$$

8. Prove or give a counterexample to the following claim:

Claim. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F},$$ and let $$T \in \cal{L}$$$$(V)$$ be a linear operator on $$V$$ . If the matrix for $$T$$ with respect to some basis on $$V$$ has all zeros on the diagonal, then $$T$$ is not invertible.

9. Prove or give a counterexample to the following claim:

Claim. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F},$$ and let $$T \in \cal{L}$$$$(V)$$ be a linear
operator on $$V$$ . If the matrix for $$T$$ with respect to some basis on $$V$$ has all non-zero elements on the diagonal, then $$T$$ is invertible.

10. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F},$$ and let $$S, T \in \cal{L}$$$$(V)$$ be linear operators on $$V$$ . Suppose that $$T$$ has $$dim(V)$$ distinct eigenvalues and that, given any eigenvector $$v \in V$$ for $$T$$ associated to some eigenvalue $$\lambda \in \mathbb{F},$$ $$v$$ is also an eigenvector
for $$S$$ associated to some (possibly distinct) eigenvalue $$\mu \in \mathbb{F}.$$ Prove that $$T \circ S = S \circ T .$$

11. Let $$V$$ be a ﬁnite-dimensional vector space over $$\mathbb{F},$$ and suppose that the linear operator $$P \in \cal{L}$$$$(V)$$ has the property that $$P^2 = P.$$ Prove that $$V = null(P ) \oplus range(P ).$$

12. (a) Let $$a, b, c, d \in \mathbb{F}$$ and consider the system of equations given by

$ax_1 + bx_2 = 0 \\ cx_1 + dx_2 = 0.$

Note that $$x_1 = x_2 = 0$$ is a solution for any choice of $$a, b, c,$$ and $$d$$. Prove that this system of equations has a non-trivial solution if and only if $$ad − bc = 0.$$

(b) Let $$A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in \mathbb{F}^{2\times2}$$ , and recall that we can deﬁne a linear operator $$T \in \cal{L}(\mathbb{F}^2 )$$ on $$\mathbb{F}^2$$ by setting $$T (v) = Av$$ for each $$v = \left[ \begin{array}{c} v_1 \\ v_2 \end{array} \right] \in \mathbb{F}^2.$$

Show that the eigenvalues for $$T$$ are exactly the $$\lambda \in \mathbb{F}$$ for which $$p(\lambda) = 0,$$ where $$p(z) = (a − z)(d − z) − bc.$$

Hint: Write the eigenvalue equation $$Av = \lambda v$$ as $$(A − \lambda I)v = 0$$ and use the ﬁrst part.

## Contributors

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