Consider the real inner product space $M_{2,2}$ $\mathbb{R}$ (The space of 2x2 matrices with real entries), with the inner product as follows:
(a) Find an orthonormal basis for the subspace U of $M_{2,2}$ $\mathbb{R}$ spanned by:
(b) Find the orthogonal projection of the matrix
M = \begin{pmatrix} 1 & 6\\ 3 & 2 \end{pmatrix}
onto U.
I honestly have no idea where to start with this question. Any assistance would be extremely appreciated.
Guide:
The purpose of Gram-Schmidt process is meant to find an orthonormal basis.
Follow the process, when the algorithm involves inner product, use the inner product that your question provided.