Problem 4. We admit the following theorems. Theorem 3 (Standard matrix of a linear transformation). Let T : R" - R" be a linear transformation. Then T is a matrix transformation, i.e., T = TA, where A E Mmn is the standard matrix of the linear transformation T and is given by A = [T(e1) . .. T(en)], i.e., the matrix with columns consisting of the image through the transformation of the standard basis vectors of R". Theorem 4 (Orthogonal projection onto a subspace). Let W C Rm be a subspace of Rm and A E Mmn have columns forming a basis for W. Then, Vv E R", the orthogonal projection of v onto W is projw(v) = A(ATA)-'ATv. The linear transformation P : Rm - R" that projects R" onto W has A(ATA) AT as its standard matrix. (a) Why is AT A invertible? (b) Let W - span 1 (;)} and v = Find the standard matrix of the orthogonal projection onto the subspace W and use this matrix to find the orthogonal projection of v onto W. (c) Same question as above with W/ - span 1 (" ). (;) . (8) } and - - C