So I've to find the eigenvalues and a basis for each corresponding eigenspaces.

Matrix A=[(2,0,0), (1,-1,0), (-1,3,3)]

After finding the determinant, I got(λ-2) (λ+1) (λ-3)=0. So λ = 2, -1, 3 are the eigenvalues.

[(λ-2, 0,0),(-1, λ+1, 0),(1,-3,λ-3)]

For λ=2; -v_2+v_3=0

3v_2-3v_3=0

-v_3=0

Let v_3 = t, then v_2=-t, but what is v_1? Is it 0? So the span is [0, -1, 1]?

For λ=-1; -3v_1-v_2+v_3=0

-3v_3=0

-4v_3=0

Let v_3=t, then what is v_2 and v_1? Are they both 0? So the span is [0,0,1]?

For λ=3; v_1-v_2+v_3=0

4v_2-3v_3=0

Let v_3=t, then v_2=(3/4)t and v_1=(-1/4)t. So the span is [(-1/4), (3/4), 1]? I think I got this right.