Define the eigenspace \(E_\lambda\) of a matrix \(A\).

Why does an eigenspace include \(\mathbf0\)?

Find \(E_2\) for \(A=\begin{pmatrix}2&0\\0&5\end{pmatrix}\).

Find \(E_5\) for \(A=\begin{pmatrix}2&0\\0&5\end{pmatrix}\).

Find the eigenspace for \(\lambda=3\) when \(A=\begin{pmatrix}3&1\\0&3\end{pmatrix}\).

Find \(E_1\) for \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\).

Find \(E_3\) for \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\).

What is \(\dim E_4\) if \(E_4=\operatorname{span}\{(1,2,0),(0,1,1)\}\) and the two spanning vectors are independent?

Find \(E_2\) for \(A=\begin{pmatrix}2&0&0\\0&2&0\\0&0&5\end{pmatrix}\).

Find \(E_0\) for \(A=\begin{pmatrix}1&-1\\2&-2\end{pmatrix}\).

Explain why an eigenspace is a subspace.

Explain why eigenspaces for distinct eigenvalues meet only in \(\mathbf0\).

For which \(a\) does \(A=\begin{pmatrix}a&0\\0&3\end{pmatrix}\) have \(E_3=\mathbb R^2\)?

For which \(k\) does \(E_1\) of \(A=\begin{pmatrix}1&k\\0&1\end{pmatrix}\) have dimension \(2\)?

Find \(E_2\) for \(A=\begin{pmatrix}2&1&0\\0&2&0\\0&0&3\end{pmatrix}\).

Find \(E_3\) for \(A=\begin{pmatrix}2&0&0\\0&3&1\\0&0&3\end{pmatrix}\).

A matrix has \(E_2=\operatorname{span}\{(1,0),(0,1)\}\). What does the matrix do to every vector in \(\mathbb R^2\)?

A student lists \(E_4=\{(1,0),(2,0),(3,0)\}\). Explain why this is not a correct eigenspace description.

Prove that \(E_\lambda=\ker(A-\lambda I)\).

Explain why the geometric multiplicity of an eigenvalue cannot exceed its algebraic multiplicity in the examples studied.