# Vector Span problem?

## Show that (10, 9, 8) lies in the vector space Span{(−1, 3, 1), (4, 1, 2)}.

Jul 7, 2018

Please see the Explanation for a Proof.

#### Explanation:

In order to show that $\vec{w} = \left(10 , 9 , 8\right) \in S P \left\{\vec{u} , \vec{v}\right\} ,$

$\vec{u} = \left(- 1 , 3 , 1\right) , \mathmr{and} , \vec{v} = \left(4 , 1 , 2\right)$, we must show that

$\exists a , b \in \mathbb{R} , s . t . , \vec{w} = a \vec{u} + b \vec{v}$.

$\therefore \left(10 , 9 , 8\right) = a \left(- 1 , 3 , 1\right) + b \left(4 , 1 , 2\right) , i . e . ,$

:. -a+4b=10...(1), 3a+b=9...(2), &, a+2b=8...(3).

Solving (1) and (3), b=3, &, a=2.

This also satisfy $\left(2\right)$.

Thus, we have, $\vec{w} = 2 \vec{u} + 3 \vec{v}$.

$\Rightarrow \vec{w} = \left(10 , 9 , 8\right) \in S p \left\{\begin{matrix}- 1 & 3 & 1 \\ 4 & 1 & 2\end{matrix}\right\}$.