signokodo7h
2021-12-03
Answered

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

Answered 2021-12-04
Author has **15** answers

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

$$S=\{\left[\begin{array}{c}4\\ 0\\ 1\\ -2\end{array}\right],\left[\begin{array}{c}3\\ 0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right]\}$$

is a set that spans W.

$$S=\{\left[\begin{array}{c}4\\ 0\\ 1\\ -2\end{array}\right],\left[\begin{array}{c}3\\ 0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right]\}$$

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