RECENT ASSIGNMENT

1. Let y — (1,2,3). In this question, you will project y on to some other vectors in R3. Show all your work and explain the major steps along the way.
(a) Let a = (0,0,1) and b = (0,1,0). Find two scalar projections, acl and ab, say, of y on a or on b and then two vector projections, say pa and pb, of y on a or on b. Explain, using the definition of the space spanned by a vector, why pa e span(a) and pb e span(b).
(b) Recall that the “error” in projecting y onto a or b is perpendicular to the space spanned by those vectors. Show that ua := y — pa is orthogonal to a and that ub := y — pb is orthogonal to b. Discuss how you can recognize this by comparing ua and ub to span(a) and span(b).
(c) Are a and b linearly independent or dependent, and how can you tell? LetX = [a, b] G R3x2 (that is, with the vectors as columns of X). What is the rank of X, and what is the space spanned by the columns ofX?
(d) The equivalent of the scalar projection of y on the pair of vectors in X is
/3 = (XTX)-1XTy.
Compute this “scalar” projection (it’s really sort of a vector of scalar projections). Then, the equivalent of a vector projection of y on the pair of vectors in X is
Compute this projection.
(e) Is px in the space spanned by the columns of X that you found two parts ago? Verify if it is or show that it is not.
(f) Finally, compute ux = y — Px and show that ux is orthogonal to both vectors in X.