Question: Find a basis for the subspace of $\mathbb{R}^4$ spanned by the given vectors:
$(-1,1,-2,0),(3,3,6,0),(9,0,0,3)$
The solution to this problem is the basis for the row space of these vectors. Is the basis for row space and subspace the same thing? Also, when solving this problem, the solution manual solved it by placing the vectors like this:
$\begin{bmatrix}-1& 1& -2& 0\\3& 3& 6& 0\\9& 0& 0& 3\end{bmatrix}$
But when finding a basis, shouldn't the vectors be placed as columns of a matrix?
Thanks!
Best Answer
The row space of a matrix is the subspace spanned by the row vectors of that matrix. Hence, if we wish to find a basis for the subspace spanned by those vectors, then we need to make those vectors rows of a matrix and find its row space.
Alternately, we could put the vectors as columns in a matrix, but then we'd need to find the column space instead. Remember that the column space of a matrix is the subspace spanned by its column vectors.