I have two subspaces $U=x_1+2x_2+x_3-3x_4+x_5=0$ and $V=\text{span}((1, 2, 0, 1, -1)^t, (-1, 0, 3, 2, 0)^t, (1, 0, 0, 0, 1)^t, (0, 2, -3, -1, -3))$
I rewrote $U$ as $ \begin{pmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{pmatrix} = \begin{pmatrix} -2 & -1 & 3 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x_2 \\ x_3 \\ x_4 \\ x_5 \end{pmatrix} $.
I also figured out that the last vector of $V$ is linearly dependent on the other three, so only the first three are necessary for the span, I put this into a matrix and performed row reduction:
$\begin{pmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{pmatrix}= \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \lambda_1 \\ \lambda_2 \\ \lambda_3 \\ \lambda_4 \end{pmatrix}$
Then I tried inserting the vectors of $V$ into the equation for $U$ like so: $(\lambda_1 + \lambda_4)+2(\lambda_2-\lambda_4)+(\lambda_3-2\lambda_4)-3(0)+0=0$. I then rewrote as $\begin{pmatrix} \lambda_1 \\ \lambda_2 \\ \lambda_3 \\ \lambda_4 \\ \end{pmatrix} = \lambda_2 \begin{pmatrix} -2 \\ 1 \\ 0 \\ 0 \end{pmatrix} + \lambda_3 \begin{pmatrix} -1 \\ 0 \\ 1\\ 0 \end{pmatrix} + \lambda_4 \begin{pmatrix} 3 \\ 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -2 & -1 & 3 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \lambda_2 \\ \lambda_3 \\ \lambda_4 \end{pmatrix}$
And by rewriting into row-echelon form, I found that all three columns together form a basis for $U\cap V$.
But using this suggestion, I found that the dimension of the intersection is 2, ( I put the first four (all of them) columns of the matrix in the rewritten $U$ form and the first three columns of the rewritten $V$ matrix next to each other and performed some row reduction) and found five pivot elements and two non-pivot variables, so I concluded that the dimension is two, in contradiction with all of the above.
So my question is, is my thinking incorrect in that I can put the vectors of one subspace into the equation of another, or am I making a mistake here, or did I not correctly apply the suggested answer link?
