need fast help will rate fast Let A 05 05 05 05 05 05 05 05

need fast help will rate fast

Let A = [0.5 -0.5 0.5 -0.5 0.5 0.5 0.5 0.5]. Note that the columns of A are orthonormal (why?). Solve the least squares problem Ax = b where b = [3 -3 3 2]. Find the projection matrix P that projects vectors in R^4 onto R(A) Compute Ax and Pb

Solution

In an mxn matrix, the columns are said to be orthonormal if A(transpose) A= I;
Here A^TA comes out to be a 2x2 matrix which is = I so we can say that the columns are orthonormal.

For part A) if you take matrix X as a 2x1 column matrix with elements x and y in it, you get: (for the elements in first 2 rows)
x-y / 2= 3
x-y/ 2 = -3
Which is not possible;