Math 6420 • PDE

Let \(x\) and \(y\) be vectors in an inner product space.
\begin{enumerate}
\item[(a)] Verify that the choice
\[
  \alpha=\frac{\langle x,y\rangle}{\lVert y\rVert^2}
\]
makes \(\lVert x-\alpha y\rVert^2\) as small as possible.
\item[(b)] Show that \(|\langle x,y\rangle|^2=\lVert x\rVert^2\lVert y\rVert^2\) if and only
if \(x\) and \(y\) are linearly dependent.
\end{enumerate}