I need to find the basis of the eigenspace for the matrix A for the eigenvalue of 4
$$A = \begin{bmatrix} 1 & 3 & 3\\ 3 & 1 &-3\\ -3 & 3& 7\\ \end{bmatrix}$$
So I subtracted 4 from the diagonals to get
$$ \begin{bmatrix} -3 & 3 & 3\\ 3 &-3 &-3\\ -3& 3 & 3 \end{bmatrix}$$
Using the third row to clear the other rows I get
$$\begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ -3 & 3 & 3\end{bmatrix}$$
Which would get $[1 \:\:\: -1 \:\:\: -1]$ for the last row. (after dividing it by -3) So $X_1 - X_2 - X_3 = 0$ thus $X_1 = X_2 + X_3$ and we get
$$X_2 \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \text{ and } X_3 \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$$
So the eigenvectors and basis should be $$ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$$
However I get back the answer
$$ \begin{bmatrix} 1 \\ 0 \\ 1\end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}$$
on my math program.
Can someone explain to me where I went wrong? I managed to answer other exercises just fine, but I seem to be getting more and more of these exercises where I get slightly different answers than those that I have found.
You are both correct. The eigenspace is a plane, and a basis is any two vectors in that plane that are linearly independent.