Okay, here's the problem:
Suppose that {$u_1,u_2,u_3$} forms a basis for a vector space V. Show that if $v_1=u_1+2u_3$$v_2=u_1+2u_2+3u_3$$v_3=u_2-u_3$ then {$v_1,v_2,v_3$} forms a basis for V.
So what I've decided so far is that if $M=\begin{bmatrix} 1&1&0\\ 0&2&1\\ 2&3&-1 \end{bmatrix}$ and $U=\begin{bmatrix} u_1&u_2&u_3 \end{bmatrix}$ then the vectors $v_1,v_2,v_3$ are the columns of $ UM$. So I just need to show, I guess, that the columns of $UM$ are linearly independent. $M$ is invertible; I'm pretty sure that's important. If I had a theorem that said that the product of two matrices with linearly independent columns has linearly independent columns then I'd be set, but I don't believe I do (I would if I knew that U was square). Am I going in a good direction here, or is there a smarter way to do this?