Orthogonal complements

Let $W \subset R^{5}$ be the space spanned by the vectors

{[\begin{matrix} 1 \\ - 2 \\ 0 \\ 2 \\ 1 \end{matrix}], [\begin{matrix} - 2 \\ 4 \\ - 1 \\ 1 \\ 2 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 2 \\ - 2 \\ 1 \end{matrix}]} .

(a) Compute the dimension of $W$ .
(b) Let $W^{⊥} = {v \in R^{5} ∣ v \cdot w = 0 for all w \in W}$ . Determine the dimension of $W^{⊥}$ , and explain how this following immediately from (a) using a theorem.
(c) Find a basis for $W^{⊥}$ .

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Let$W\subset {\bf R}^5$ be the space spanned by the vectors
\[
	\left\{\begin{bmatrix} 1 \\ -2 \\ 0 \\ 2 \\ 1\end{bmatrix},\begin{bmatrix} -2 \\ 4 \\ -1 \\ 1 \\ 2\end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 2 \\ -2 \\1\end{bmatrix}\right\}.
\]
\begin{enumerate}[label=\alph*)]
	\item Compute the dimension of $W$.
	\item Let $W^{\perp}=\{{\bf v}\in {\bf R}^5\,\mid\, {\bf v}\cdot {\bf w}=0\text{ for all }w\in W\}$. Determine the dimension of $W^{\perp}$, and explain how this following immediately from (a) using a theorem.
	\item Find a basis for $W^{\perp}$.
\end{enumerate}