# 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.

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.

• Thanks Gerry Myerson. Row space isn't discussed until the next section. Why are we supposed to take the transpose and then row reduce? – BCLC Sep 27 '16 at 0:46
• The span of the rows of a matrix is invariant under elementary row operations, so it makes sense to take the vectors as rows in a matrix and then row reduce. – Gerry Myerson Sep 27 '16 at 2:28
• So, are we OK now? – Gerry Myerson Sep 28 '16 at 6:31
• "thinking about how to explain it otherwise." Thinking about how to explain what, exactly? – Gerry Myerson Sep 28 '16 at 7:17
• That question presupposes that there's something special about making the given vectors the columns of the matrix. But if you start by making the vectors the rows of the matrix, the question of the transpose never arises. – Gerry Myerson Sep 28 '16 at 9:13