I was wondering if someone could explain the difference between an eigenspace and a basis of an eigenspace. I only somewhat understand the latter.
Let's say that the row reduced form for a matrix with $\lambda=-1$ is:
$\begin{bmatrix} 1 & 1 & 1 & | & 0 \\ 0 & 0 & 0 & | & 0 \\ 0 & 0 & 0 & | & 0 \\ \end{bmatrix}$
The bases of the eigenspace would then be, if I understood correctly,
$E_{\lambda=-1}=\{\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}=a\begin{bmatrix}-1\\1\\0\end{bmatrix}+b\begin{bmatrix}-1\\0\\1\end{bmatrix} : a,b \in \mathbb{R} \}$
But what is the eigenspace?
Best Answer
This is actually the eigenspace:
$$E_{\lambda=-1}=\left\{\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}=a_1\begin{bmatrix}-1\\1\\0\end{bmatrix}+a_2\begin{bmatrix}-1\\0\\1\end{bmatrix} : a_1,a_2 \in \mathbb{R} \right\}$$
which is a set of vectors satisfying certain criteria.
The basis of it is:
$$\left\{\begin{pmatrix}-1\\1\\0\end{pmatrix}, \begin{pmatrix}-1\\0\\1\end{pmatrix}\right\}$$
which is the set of linearly independent vectors that span the whole eigenspace.