I just need someone to check if I'm doing this properly.
Find a basis for: $span\{(1,1,1),(1,0,1),(0,0,0),(0,1,0)\}$
I formed matrix A using these 4 vectors as columns;
$A=\begin{bmatrix}1&1&0&0\\1&0&0&1\\1&1&0&0\end{bmatrix}^T$
$rref = \begin{bmatrix}1&0&0&-1\\0&1&0&1\\0&0&0&0\end{bmatrix}^T$
I'm a bit uncertain about the next step. I have pivots in the first two columns, meaning they are linearly independent, so does that mean my basis is just the first two columns of $A$?
$basis=\{ (1,1,1), (1,0,1) \}$
Best Answer
The span is all linear combinations of the basis vectors.
$(1,1,1) = (1,0,1) + (0,1,0)$ and $(0,0,0) = 0*(1,0,1)$ so we can remove these two vectors from a potential basis.
$\alpha(1,0,1) + \beta(0,1,0) = (\alpha,\beta,\alpha) = (0,0,0)$ only when $\alpha = \beta = 0$.
Thus $(1,0,1)$ and $(0,1,0)$ are linearly independent and thus a basis of $\text{span}\{(1,1,1),(1,0,1),(0,0,0),(0,1,0)\}$.