Bases for a subspace and its orthogonal complement
Let
- Find (with justification) a basis for
. - Find (with justification) a basis for
, the subspace of orthogonal to under the usual dot product.
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Let $V\subset {\bf R}^5$ be the subspace defined by the equation
\[
x_1-2x_2+3x_3-4x_4+5x_5=0.
\]
\begin{enumerate}[label=\alph*)]
\item Find (with justification) a basis for $V$.
\item Find (with justification) a basis for $V^{\perp}$, the subspace of ${\bf R}^5$ orthogonal to $V$ under the usual dot product.
\end{enumerate}