Are eigenspaces unique?

Question

I have calculated an eigenspace of a matrix. It is 2 dimensional. I checked it with WolframAlpha, but in WolframAlpha's solution one basis vector in this eigenspace is different from my solution.

Answer 1

Hint. Doing it by hand, you can check that the two spaces are equal as in the following example. Let $$V={\rm span}\{(1,2,0),(3,5,1)\}\quad\hbox{and}\quad W={\rm span}\{(7,8,6),(3,-4,10)\}\ .$$ Set up an augmented matrix and row-reduce: $$\pmatrix{1&3&|&7&3\cr 2&5&|&8&-4\cr 0&1&|&6&10} \sim\pmatrix{1&3&|&7&3\cr 0&-1&|&-6&-10\cr 0&0&|&0&0\cr}\ .$$ Considering each of the two right hand sides separately, each has a solution as a linear combination of the vectors on the left hand side. That is, $(7,8,6)$ and $(3,-4,10)$ are both linear combinations of $(1,2,0)$ and $(3,5,1)$. So $W$ is a subspace of $V$. You can now do the same sort of calculation "in the other direction"; more simply, since $W$ and $V$ have the same dimension they must be equal.