Orthogonal complements
Let
- Compute the dimension of
. - Let
. Determine the dimension of , and explain how this following immediately from (a) using a theorem. - Find a basis for
.
View code
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}