a (a) In step 1, set q1 = "1 Argue that (11 is a unit vector and lll' Span{q1} = Span{a1}. (b) In Step 2, we want to nd a unit vector q; orthogonal to ql and in Span{a1 , 3.2} . (i) Find the matrix P1 projecting to (Span{q1})i. (ii) Compute a; = P132 (iii) Set q2 = a2 Argue that q2 is a unit vector orthogonal to q1 and in Span { a1, a2). (c) Write down the jth step of this algorithm based on what you have seen so far: (i) What is the formula for Pi-1, the matrix projecting to (Span {q1, . . ., qj-1})? (ii) Compute a, = Pj-laj (iii) Set qi = Jaff. Argue that q; is a unit vector orthogonal to Span { q1, . .., qj-1} and in Span {a1, ..., aj}. (d) Use the above algorithm to orthonormalize the basis consisting ofInput: A basis a1, . .., an of Rn. Output: An orthonormal basis q1, . .., an of In