# Prove that these two sets span the same subspace - Why take the transpose?

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.

1. Why are the transposes row reduced instead?

2. 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.

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 ${\bf R}^3$.