# Let W be the set of all vectors of the form shown, where a, b, and c r

Let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or give an example to show that W is not a vector space.
$\left[\begin{array}{c}4a+3b\\ 0\\ a+b+c\\ c-2a\end{array}\right]$
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Dona Hall

Since any vector w in W may be written as
$w=a\left[\begin{array}{c}4\\ 0\\ 1\\ -2\end{array}\right]+b\left[\begin{array}{c}3\\ 0\\ 1\\ 0\end{array}\right]+c\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right]$
We see that
$S=\left\{\left[\begin{array}{c}4\\ 0\\ 1\\ -2\end{array}\right],\left[\begin{array}{c}3\\ 0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right]\right\}$
is a set that spans W.
$S=\left\{\left[\begin{array}{c}4\\ 0\\ 1\\ -2\end{array}\right],\left[\begin{array}{c}3\\ 0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right]\right\}$