studybyyourself

Question 1

Let $i$ be a linear transformation. What are its eigenvalues ?

$\begin{equation*} i \colon \, \mathbb{R}^2 \to \mathbb{R}^2 \\ \quad \vec{x} \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \vec{x} \end{equation*}$

infinitely many eigenvalues

$\{-1,1\}$

$\{1\}$

$\{0\}$

Question 2

Let $f$ be a linear transformation. What are its eigenvectors ?

$\begin{equation*} f \colon \, \mathbb{R}^2 \to \mathbb{R}^2 \\ \quad \vec{x} \mapsto \begin{pmatrix} -1 & -4 \\ -2 & 1 \end{pmatrix} \vec{x} \end{equation*}$

$\{\}$

$\big\{ \left(\begin{smallmatrix} 1 \\ 3 \end{smallmatrix} \right), \left( \begin{smallmatrix} 2 \\ 5 \end{smallmatrix} \right) \big\}$

infinitely many eigenvectors

$\big\{ \left( \begin{smallmatrix} 1 \\ 2 \end{smallmatrix} \right), \left( \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \right)\big\}$

Question 3

Why must the associated matrix to a linear transformation be square for this latter to have eigenvectors ?

because the columns of the matrix must be linearly independent

because $\vec{x}$ and $\vec{y}$ must have the same size

because the linear transformation must be bijective

because an eigenvector must be different than the null vector

11.1

Question 1

Question 2

Question 3