#5 Please

The following preamble applies to both problems 4 and 5: Let W C R" be a subspace and let WA C R" be its orthogonal complement. Let S = fun,...; ux} be an orthogonal basis for W and S' = {v1, ..., v} be an othogonal basis for WI. The goal of these exercises is to give two different arguments for the fact that (* ) dim W + dim WI = n. 4. Show that SU S' is an orthogonal basis for R" and deduce (*). 5. Let p = i o projw : R" -> W R" be the function defined by orthogonal projection onto W followed by inclusion, i.c. i(x) = x for all x ( W. Show that the null space of p Null(p) = {ve R" : p(v) =0} is exactly W. Now explain why and how one can deduce (*) from this result and the rank theorem