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 $\lambda$.

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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 $V$ 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 $a_1=2, a_2=7$.

  4. 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.

  5. 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 $a_1, a_2$.