Need help understanding the span of the following vectors.

Question

$$ \begin{matrix} 1 & 0 & -2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \\ \end{matrix} $$

I am told that the span of vectors equal $R^m$ where $m$ is the rows which has a pivot in it. So when describing the span of the above vectors, is it correct it saying that they don't span $R^3$ but only span $R^2$?.

Thanks

Answer 1

Yes, once your matrix is in the pivot form, that is one way to calculate the dimension of the row space. In this case yes you have 2 pivots so the dimension would be 2. I would be careful in saying they span $R^2$. Since though $R^2$ and $\{(x,y,0),x\in R, y\in R\}$ are similar, they are not exactly the same. This is because from your perspective you would call $\{(x,y,0),x\in R, y\in R\}$ and $\{(x,0,y),x\in R, y\in R\}$ both $R^2$, but they are in fact different planes in $R^3$

Answer 2

Since your vectors are linearly dependent but two of them are linearly independent, they will span a two dimensional subspace of $R^3$

The span is not necessarily $R^2$ but it is a plane passing through the origin.