# Eigenvalues and Eigenvectors of Linear Transformations

## Eigenvalues and Eigenvectors of Linear Transformations

Definition

Let $T:V \to V$ be a linear transformation from a vector space $V$ to itself.

1. We say that $\lambda$ is an eigenvalue of $T$ if there exists a nonzero vector $\mathbf{v}\in V$ such that $T(\mathbf{v})=\lambda \mathbf{v}$.
2. For each eigenvalue $\lambda$ of $T$, nonzero vectors $\mathbf{v}$ satisfying $T(\mathbf{v})=\lambda \mathbf{v}$ is called eigenvectors corresponding to $\lambda$.

=solution

### Problems

1. Let $T$ be the linear transformation from the vector space $\R^2$ to $\R^2$ itself given by
$T\left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right)= \begin{bmatrix} 3x_1+x_2 \\ x_1+3x_2 \end{bmatrix}.$ (a) Verify that the vectors
$\mathbf{v}_1=\begin{bmatrix} 1 \\ -1 \end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ are eigenvectors of the linear transformation $T$, and conclude that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis of $\R^2$ consisting of eigenvectors.
(b) Find the matrix of $T$ with respect to the basis $B=\{\mathbf{v}_1, \mathbf{v}_2\}$.

2. Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by $T(ax+b)=(3a+b)x+a+3$ for any $ax+b\in P_1$.
(a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation $T$.
(b) Find a basis $B’$ of the vector space $P_1$ such that the matrix of $T$ with respect to $B’$ is a diagonal matrix.
(c) Express $f(x)=5x+3$ as a linear combination of basis vectors of $B’$.

3. Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by $T( ax^2 + bx + c ) = 2ax + b$. Is $T$ diagonalizable? If so, find a diagonal matrix which represents $T$. If not, explain why not.
4. Let $V$ be a real vector space of all real sequences $(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots)$. Let $U$ be a subspace of $V$ defined by
$U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.$ Let $T$ be the linear transformation from $U$ to $U$ defined by
$T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots).$ (a) Find the eigenvalues and eigenvectors of the linear transformation $T$.
(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying $a_1=2, a_2=7$.

5. Let $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$. Prove that the following two statements are equivalent.
(a) There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through the origin that are mapped onto themselves: $T(L_1)=L_1 \text{ and } T(L_2)=L_2$.
(b) The matrix $A$ has two distinct nonzero real eigenvalues.

6. Let $V$ be a real vector space of all real sequences $(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots)$.
Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ for $k=1, 2, \dots$. Let $T$ be the linear transformation from $U$ to $U$ defined by
$T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots).$ Let $B=\{\mathbf{u}_1, \mathbf{u}_2\}$ be a basis of $U$, where
\begin{align*}
\mathbf{u}_1&=(1, 0, -3, -15, -66, \dots)\\
\mathbf{u}_2&=(0, 1, 5, 22, 95, \dots).
\end{align*}
Let $A$ be the matrix representation of the linear transformation $T: U \to U$ with respect to the basis $B$.
(a) Find the eigenvalues and eigenvectors of $T$.
(b) Use the result of (a), find a sequence $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and the initial condition $a_1=1, a_2=1$.
(c) Find the formula for the sequences $(a_i)_{i=1}^{\infty}$ satisfying the linear recurrence relation $a_{k+2}-5a_{k+1}+3a_{k}=0$ and express it using $a_1, a_2$.