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My textbook states the following:

Eigenspaces for nonzero eigenvalues are subspaces of the column space of the matrix while the 0-eigenspace is the null space of the matrix. Geometrically, they are lines or planes through the origin in $\mathbb{R}^3$ if $A$ is a $3 \times 3$ matrix (other than the identity matrix).

Why does it say, "other than the identity matrix"? What does it mean when it says this?

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I think he means that in the case of the identity matrix, all of $\mathbb{R}^3$ is an eigenspace (which is neither a line nor a plane, but more than those). This only happens if the matrix is the identity or a multiple of it.

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    I still do not understand why all of $\mathbb{R}^3$ is an eigenspace in the case of the identity matrix? For instance, if I have $A = I_3$, $v = (2, 3, 4)$, and $\lambda (eigenvalue) = -1$, then how does this correspond to what they're saying? – 2017-02-20
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    The only eigenvalue of the identity matrix is $\lambda = 1$. All of $\mathbb{R}^3$ is an eigenspace (associated to $\lambda=1$) because each $v \in \mathbb{R}^3$ satisfies $Av=\lambda v$. – 2017-02-20
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    You can't have that situation though as all the eigenvalues of the identity matrix are 1. In the case of the identity, the nullspace is exactly the zero vector while the eigenspace spans all of \mathbb{R}^3$ – 2017-02-20
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    You're right. I understand now. Thank you. – 2017-02-20