Larson Edwards Falvo - Elementary Linear Algebra
For 51, I was thinking that we had to show that
$$\{c_1(1,2,-1)^T + c_2(0,1,1)^T + c_3(2,5,-1)^T\} = \{c_1(-2,-6,0)+c_2(1,1,-2)\}$$
So I wanted to row reduce
$$\begin{bmatrix} 1 & 0 & 2\\ 2 & 1 & 5\\ -1 & 1 & -1 \end{bmatrix}$$
and
$$\begin{bmatrix} -2 & 1\\ -6 & 1\\ 0 & -2 \end{bmatrix}$$
much like in 52.
Why are the transposes row reduced instead?
Also, if I reduce the matrices above then I get
$$\begin{bmatrix} 1 & 0 & 2\\ 0 & 1 & 1\\ 0 & 0 & 0 \end{bmatrix}$$
and
$$\begin{bmatrix} 1 & 0\\ 0 & 1\\ 0 & 0 \end{bmatrix}$$
Would it follow that $S_1$ and $S_2$ span the same subspace? It looks like they are both in reduced row echelon form although not in reduced column echelon form.
The row space is invariant under elementary row operations, so for #51 it makes sense to take the vectors as the rows in a matrix, and bring to reduced row-echelon form.
I would have used rows for #52, also, but since the reduced form turns out to be the identity matrix, you can conclude that in each case the vectors span ${\bf R}^3$.
The computation you've done, I'm not sure it proves anything, other than that the two spaces both have dimension 2.