how to find the span of 3 vectors

Question

how do I find the span of 3 vectors: specifically for $(1, 1, 2)$; $(0, -1, 1)$;$(2, 5, 1)$ ? I know the answer is $(a, b, 3a-b)$. I just don't know how you get to that answer. Thanks in advance

Answer 1

Span means the set of vectors which can be obtained as a linear combination of the given vectors. Let $\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix} \in \text{Span} (S)$, where $S=\{(1,1,2), \, (0,-1,1), \, (2,5,1)\}$. Then $$x\begin{bmatrix}1\\1\\2\end{bmatrix}+y\begin{bmatrix}0\\-1\\1\end{bmatrix}+z\begin{bmatrix}2\\5\\1\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}.$$ You need to find the condition on $b_1,b_2,b_3$ for which this system of equation will be consistent. For that consider the augmented matrix \begin{align*} \left[\begin{array}{rrr|r} 1&0&2 &b_1\\ 1&-1&5 &b_2\\ 2&1&1&b_3 \end{array} \right] \end{align*} After a row operation you will get \begin{align*} \left[\begin{array}{rrr|r} 1&0&2 &b_1\\ 0&-1&3 &b_2-b_1\\ 0&1&-3&b_3-2b_1 \end{array} \right] \end{align*} and then \begin{align*} \left[\begin{array}{rrr|r} 1&0&2 &b_1\\ 0&-1&3 &b_2-b_1\\ 0&0&0&b_2+b_3-3b_1 \end{array} \right] \end{align*} I hope you can take it from here.