Let V be a vector space with basis $u_1, u_2, u_3$ and let $T : V → V$ be the linear transformation such that
$T(u_1)=u_1 −u_2 −u_3$
$T(u_3)=−u_1 +3u_2 −5u_3$
Find bases for the null space and the image of T and determine the rank and nullity of T .
I think I solved for the null space and nullity first.
After solving the system of equations with the solutions equal to $0$, I came up with $u_3=k$ which gave $u_1=4k$ and $u_2=3k$ which gives $(4k,3k,k)$ nullity 3. Was this the correct method (gaussian elimination)?
And for solving for the basis of the image would I use the same method but instead of having $0$ in the right-most column for the solutions I would have 1,2,-5 (in descending order) corresponding to the coefficient in the given equation.
*After a second look I'm almost sure this is incorrect since the coefficients aren't the same as the T($u_n$)