Eigen space and column space of matrix A is same?

Question

Let we have $n \times n$ matrix A with full rank.

Eigen vectors of A is in the column space of A, because eigen vector is made by linear combination of column vectors of A.

And, eigen vector also span the eigen space with same dimenson of column space of A.

So, I wonder whether the column space of A is equal to eigen space of A.

Answer 1

In general, no. Take, for instance, $A=\left[\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right]$. It has full rank, but the column space is $\mathbb R^2$, whereas the only eigenspace is $\{(x,0)\,|\,x\in\mathbb R\}$.