What should I be denoting the "solution" to a homogeneous system?

Question

Suppose I have the RREF of matrix A (which is a homogeneous system ie A is technically augmented with a zero column but I didn't add it): $$ \begin{bmatrix} 1&0&-1&0&-3\\ 0&1&3&0&-5\\ 0&0&0&1&-4\\ 0&0&0&0&0\\ \end{bmatrix} $$ And I solve for the variables noting that $x_3, x_5$ are free variables. In finding the bases for the solution I wrote, instinctively $$ =s \begin{bmatrix}1\\-3\\1\\0\\0 \end{bmatrix} +t \begin{bmatrix}3\\5\\0\\4\\1\end{bmatrix} $$ But what did I set this equal to? What should be on the other side of the equals sign other than like the words "the basis is ="

Answer 1

Well, all solutions $x$ of the linear system $Ax = 0$ are of the form $$ x = s \begin{bmatrix} 1 \\ -3 \\ 1 \\ 0 \\ 0\end{bmatrix} + t \begin{bmatrix} 3 \\ 5 \\ 0 \\ 4 \\ 1 \end{bmatrix}, \quad t, s \in \mathbb{R} $$

So, the answer is that the left side of the equality is $x$, a solution of the linear system.