# Eigenspace versus Basis of Eigenspace

I was wondering if someone could explain me the difference between eigenspace and basis of eigenspace. Right now I have only been able to 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,

$E_{\lambda=-1}=\{\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}=a_1\begin{bmatrix}-1\\1\\0\end{bmatrix}+a_2\begin{bmatrix}-1\\0\\1\end{bmatrix} : a,b \in \mathbb{R} \}$

But what is the eigenspace?

• An eigenspace is in particular a vector subspace, like your $E_{\lambda = -1}$, or more commonly, $E_{-1}$, and a basis thereof is just that. – Travis Mar 2 '15 at 11:06

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.

• Aha, I get it. Thank you very much, Kitty! – Akitirija Mar 2 '15 at 11:30