Finding the basis of a set of vectors.

Let $$S = \left(\left[\begin{array}{c} 1 \\ -1 \\ 3 \\ 2\end{array}\right], \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ 3\end{array}\right], \left[\begin{array}{c} 1 \\ 5 \\ -7 \\ 0\end{array}\right], \left[\begin{array}{c} 4 \\ -1 \\ 7 \\ 7\end{array}\right]\right)$$ and $$V =$$ span$$(S)$$. Find a basis of $$V$$.

I started by putting the vectors into the columns of a matrix. Then I row reduced the matrix and selected all the pivot columns as the vectors of the basis. However, when I checked the solutions, the vectors were placed into the matrix as rows instead of columns.

Does it matter if the vectors are placed as columns or rows?

• Row reducing the matrix of column vectors will also produce a basis, given that you take the pivot columns in the original matrix and not the row-reduced matrix. Both your method and the textbook's method will work. Commented Jun 30, 2022 at 19:43

So actually what you did was calculate a basis for the subspace spanned by $$(1,2,1,4) , (-1,1,5,-1),(3,1,-7,7),(2,3,0,7)$$ because when you applied elementary row operations on them, you took linear combinations of these vectors to get new vectors which have to lie in the span.
If you do it correctly, you should end up with a basis $$\{(3,0,4,5) ,(0,3,-5,-1)\}$$ .