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Linear Algebra

Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling

Chapter 7

Eigenvalues and Eigenvectors - all with Video Answers

Educators

Chapter Questions

01:34

Problem 1

Let $T \in \mathcal{L}\left(\mathbb{F}^{2}, \mathbb{F}^{2}\right)$ be defined by
$$T(u, v)=(v, u)
$$for every $u, v \in \mathbb{F}$. Compute the eigenvalues and associated eigenvectors for $T$.

Sarah Klein
Sarah Klein
Numerade Educator
03:58

Problem 2

Let $T \in \mathcal{L}\left(\mathbb{F}^{3}, \mathrm{~F}^{3}\right)$ be defined by
$$T(u, v, w)=(2 v, 0,5 w)$$
for every $u, v, w \in \mathbb{F}$. Compute the eigenvalues and associated eigenvectors for $T$.

Sarah Klein
Sarah Klein
Numerade Educator
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Problem 3

Let $n \in \mathbb{Z}_{+}$ be a positive integer and $T \in \mathcal{L}\left(\mathbb{F}^{n}, \mathbb{F}^{n}\right)$ be defined by
$$T\left(x_{1}, \ldots, x_{n}\right)=\left(x 1+\cdots+x_{n}, \ldots, x_{1}+\cdots+x_{n}\right)$$
for every $x_{1}, \ldots, x_{n} \in \mathbb{F}$. Compute the eigenvalues and associated eigenvectors for $T$.

Victor Salazar
Victor Salazar
Numerade Educator
01:02

Problem 4

Find eigenvalues and associated eigenvectors for the linear operators on $\mathrm{F}^{2}$ 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) $\left[\begin{array}{ll}0 & 3 \\ 4 & 0\end{array}\right]$,
(d) $\left[\begin{array}{cc}-2 & -7 \\ 1 & 2\end{array}\right]$,
(e) $\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$,
(f) $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

Raj Bala
Raj Bala
Numerade Educator
02:46

Problem 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) $\left[\begin{array}{cccc}1 & 3 & 7 & 11 \\ 0 & \frac{1}{2} & 3 & 8 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 2\end{array}\right]$

M Hassan Anwar
M Hassan Anwar
Numerade Educator
01:08

Problem 6

For each matrix $A$ below, describe the invariant subspaces for the induced linear operator Ton $\mathbb{F}^{2}$ that maps each $v \in \mathbb{F}^{2}$ to $T(v)=A v$
(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) $\left[\begin{array}{ll}2 & 3 \\ 0 & 2\end{array}\right]$,
(d) $\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]$

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 7

Let $T \in \mathcal{L}\left(\mathbb{R}^{2}\right)$ be defined by$$T\left(\begin{array}{l}
x \\y\end{array}\right)=\left(\begin{array}{c}x \\
x+y\end{array}\right), \text { for all }\left(\begin{array}{l}x \\y
\end{array}\right) \in \mathbb{R}^{2}$$
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 eigenvectors, where
$$v_{+}=\left(\begin{array}{c}1 \\\lambda_{+}\end{array}\right), v_{-}=\left(\begin{array}{c}1 \\\lambda_{-}\end{array}\right)$$
(c) Show that $\left(v_{+}, v_{-}\right)$ is a basis of $\mathbb{R}^{2}$.
(d) Find the matrix of $T$ with respect to the basis $\left(v_{+}, v_{-}\right)$ for $\mathbb{R}^{2}$ (both as the domain and the codomain of $T$; call this matrix $B$ ).

Victor Salazar
Victor Salazar
Numerade Educator