Answer :
Answer:
[tex]W=\{\left[\begin{array}{ccc}a+2b\\b\\-3a\end{array}\right]: a,b\in\mathbb{R} \}[/tex]
Observe that if the vector [tex]x=\left[\begin{array}{ccc}x\\y\\z\end{array}\right][/tex] is in W then it satisfies:
[tex]\left[\begin{array}{ccc}x\\y\\z\end{array}\right]=\left[\begin{array}{c}a+2b\\b\\-3a\end{array}\right]=a\left[\begin{array}{c}1\\0\\-3\end{array}\right]+b\left[\begin{array}{c}2\\1\\0\end{array}\right][/tex]
This means that each vector in W can be expressed as a linear combination of the vectors [tex]\left[\begin{array}{c}1\\0\\-3\end{array}\right], \left[\begin{array}{c}2\\1\\0\end{array}\right][/tex]
Also we can see that those vectors are linear independent. Then the set
[tex]\{\left[\begin{array}{c}1\\0\\-3\end{array}\right], \left[\begin{array}{c}2\\1\\0\end{array}\right]\}[/tex] is a basis for W and the dimension of W is 2.