G HW Problem 11 Solve by hand without using calculator matrix functions. Show all intermediate steps. Let v (1) = [ 3 1 1 −1 ]T , v (2) = [ 1 −2 0 1 ]T and v (3) = [ −1 1 5 3 ]T . (i) (3 pts.) Show that v (1) , v (2) and v (3) are (pairwise) orthogonal, and compute their norms. In what follows, let s = [ 1 6 2 7 ]T . (ii) (3 pts.) Determine the projection f (i) of s onto each v (i) (where i = 1, 2, 3). (iii) (2 pts.) Determine the angle between s and v (1) . (iv) (3 pts.) Determine the angle between s and the plane defined by v (2) and v (3) . (v) (3 pts.) Determine the projection g of s onto the three-dimensional subspace defined by v (1) , v (2) and v (3) . (vi) (2 pts.) Verify that error vector s − g is orthogonal to each of v (1) , v (2) and v (3) . (vii) (4 pts.) Solve the system     3 1 −1 1 1 −2 1 2 1 0 5 −2 −1 1 3 3         d1 d2 d3 d4     =     1 8 4 9     without using Gaussian elimination. (Hint: What is the relationship between s − g and the fourth column of the matrix above? )