Check if a row is in the span of a matrix
Let your vector be $x$, and let the first row be $r$.

If $x_1 = 1$ and $r_1 = 0$, then $x$ is not in the span.

If $x_1 = 0$ and $r_1 = 1$, then $r$ cannot be part in any linear combination that results in $x$, so we can remove $r$ and continue.

If $x_1 = 0$ and $r_1 = 0$, then we can remove the first column and continue.

If $x_1 = 1$ and $r_1 = 1$, then for $x$ to be a linear combination of the rows, $r$ must be one of the rows in the linear combination. So we can replace $x$ with $x \oplus r$, remove the first row and column, and continue.



You can also present this algorithm in an iterative manner.

Let $i$ be the index of the first $1$ in $x$. For $x$ to be in the span, there must be a row whose first $1$ is also at index $i$. Find this row $r$, replace $x$ with $x \oplus r$, and continue, until eventually $x = 0$.