Q:

Confusion regarding span

I am trying to understand the concept of span in linear algebra, but the more I ponder the more I get confused.
Let's say I am trying to find the span of a set in $\mathbb R^2$. The set consists of two vectors. I set up the matrix equation and apply row reduction. Row reduction leads me to an infinite solution. Now does an infinite solution imply my set doesn't span $\mathbb R^2$?

A:

Infinite solutions implies that the rank of your coefficient matrix is less than $2$. This means that the two vectors are not linearly independent and don't span $\Bbb R^2$, since you need $2$ linearly independent vectors to span $\Bbb R^2$. In general, there is no need to set up a matrix equation; you can convert the matrix into its row echelon form and observe the rank straight-away.
For example, if your two vectors are $v_1(1,2),v_2(2,4)$, it is easy to observe that $2v_1-v_2=0$. Hence, $v_1,v_2$ are linearly dependent and don't span $\Bbb R^2$. The general method for a set of vectors in $\Bbb R^n$ would involve writing them as row or column vectors of a matrix and obtaining its rank, like this:$$A=\begin{bmatrix}1&2\\2&4\end{bmatrix}\xrightarrow{R_2\to R_2-2R_1}\begin{bmatrix}1&2\\0&0\end{bmatrix}$$The rank, which gives the dimension of the subspace of $\Bbb R^2$ spaned by $v_1$ and $v_2$, is $1<2=\dim\Bbb R^2$.

