# Find the orthonormal basis for the subspace $U$ of $M_{2,2}(\mathbb{R})$ spanned by

Consider the real inner product space $$M_{2,2}(\mathbb{R})$$ (the space of 2 x 2 matrices with real entries), with inner product:

(a) Find the orthonormal basis for the subspace $$U$$ of $$M_{2,2}(\mathbb{R})$$ spanned by

I understand that to find the orthanormal basis for the subspace I have to use the Gram-Schmidt process, however, Im not quite sure how to do this. I have only every used this process with vectors and I'm not sure to do it in this case.

Now, for the orthogonal projection (part b) I am also confused. I think I am supposed to do the following: $$U = span$${S}.

Any help with these questions would be very appreciated

• Please take the time to enter those expressions as text instead of pasting pictures of them. You question should be comprehensible with images disabled. Images are neither searchable nor accessible to screen readers, nor do they show up in summaries. You can find a quick reference to formatting mathematical expressions using MathJax here. – amd Mar 26 at 5:04
• I think that you do know how to apply the Gram-Schmidt process. You just have to use the given inner product instead of the dot product that you’re no doubt used to. – amd Mar 26 at 5:05