Finding a basis for a spanning list by columns vs. by rows

From this answer, I see that, given a list of vectors, we can put them as the columns of a matrix, then reduce it, and the pivots will tell us which of the original vectors form a basis for the span of those vectors.

However, I'm wondering, can't we just put the list of vectors as the rows of a matrix, then reduce it, and find a basis this way? Since row operations don't change the row space, this method should also work. We just look at the rows that have a leading variable.

Though, I can see that the first method of putting the vectors in a column might be useful if we want to find out which of the original vectors form a basis.

Consider the matrix $$A$$ whose columns are the vectors $$v_1, \dots, v_n$$ from the $$m$$-dimensional vector space $$V.$$ We have that $$A^t$$ (the transpose of $$A$$) is the matrix whose rows are the vectors $$v_1, \dots, v_n.$$
Like you mentioned, we can determine which of the vectors $$v_1, \dots, v_n$$ are linearly independent by putting the matrix $$A$$ in reduced row-echelon form. Explicitly, the vectors $$v_j$$ for each column $$j$$ with a pivot are linearly independent, and the rest of the vectors can be written as a linear combination of the $$v_j.$$ (In fact, the coefficients of the relation of linear dependence are determined by the entries of the column of the corresponding vector. Check that for $$v_1 = (1, 2, 5)$$ and $$v_2 = (-3, -5, -13),$$ we have that $$v_4 = (2, 1, 4) = -7v_1 - 3v_2.$$ Compare this to the problem you mention.)
Like you have seen, there are other advantages of writing the vectors as the columns; however, it is true that the column rank and the row rank of any matrix are equal, hence the number of pivots of the matrix $$A^t$$ is the same as the number of pivots of the matrix $$A,$$ and one can just as legitimately row-reduce $$A^t.$$ Unfortunately, the interpretation of the coefficients of the matrix $$A^t$$ is not as clear as in the other method. (Check that the transpose of the matrix in the problem you mention is $$\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix};$$ however, the coefficients of this matrix have no immediate meaning.)