Determining span, is there an easier way to remember it?

Question

As I understand it, to set up a problem to determine if the vector spans $ \mathbb{R}^n$ or if the given vector is in the Span of $$(v_1,v_2,...,v_n)$$ you take the vector and set up an augmented matrix that is equal to the vector $w$. (I'm not quite sure the meaning of the vector $w$ as it pertains to the vector $v$). Then row reduce. After that I am lost. I'm looking for a way to simplify this concept and the steps it takes to get there.

Answer 1

Given a finite subset $V\subset\mathbb R^n$, say $V=\{v_1,\ldots,v_m\}$, a vector $b$ is in the span of $V$ if there exist scalars $c_1,\ldots,c_m$ such that $$ b = c_1v_1 + \cdots + c_mv_m.$$ This is equivalent to whether there exists a solution to $Ax=b$, where $A$ has columns $v_1, \ldots, v_m$. Note that elementary row operations do not change the column space of a matrix, so it suffices to compute the row echelon form of the augmented matrix $[A|b]$. Then $Ax=b$ has a solution if and only if there is no row of the form $$\begin{bmatrix}0&0&\cdots&0&|&b_j\end{bmatrix}$$ in the row echelon form of $[A|b]$.