How to find a basis and use rank-nullity theorem to find dim(V) of a subspace?

Question

$V = \{ (a, b, c, d) \in \mathbb{R}^4 | a - 2b + 3c + d = 0 \}$ is a subspace of $\mathbb{R}^4$. How do I use the rank-nullity theorem to find $\dim V $ & how do I find a basis?

For $\mathrm{rank}(A)+\textrm{nullity}(A) = n$ I think $n = 4$ and I know I have to find $\mathrm{nullity}(A)$ but how do I find $\mathrm{rank}(A)$?

Thank you in advance.

Answer 1

To find a basis, rewrite \begin{align} V&=\{(a,b,c,d)\in\mathbb{R}^{4}| a=2b-3c-d\}\newline &=\{(2b-3c-d,b,c,d)| b,c,d\in\mathbb{R}\}\newline &=\{b(2,1,0,0) + c(-3,0,1,0)+d(-1,0,0,1)| b,c,d\in\mathbb{R}\}. \end{align}

So a basis of $V$ is $\{(2,1,0,0), (-3,0,1,0), (-1,0,0,1)\}$.