Parallelogram Law: Prove that for any two vectors \({\bf x}\) and \({\bf y}\) with the same number of coordinates, \(||{\bf x}+{\bf y}||^2 + ||{\bf x}-{\bf y}||^2 = 2 \Bigl(||{\bf x}||^2 + ||{\bf y}||^2\Bigr).\)
Note how we showed the above using only the properties of an inner product (linearity in each argument and symmetry in arguments), without using the actual form of the matrix dot product. When you read the section about norms in general, keep this in mind. Do the next two problems similarly.
Show that if \({\bf x}\) and \({\bf y}\) are both orthogonal to \(\bf z\), then so is any linear combination of \({\bf x}\) and \({\bf y}\).
Prove Cauchy-Schwartz inequality:
\[{\bf x}^T{\bf y} \le ||{\bf x}|| \,||{\bf y}||.\]