Question

3. The projection theorem we learned in class focuses on projecting onto subspaces. In general, W, the set you are projecting onto, does not actually have to be subspace of V. (a) Let a and b be nonzero vectors in an inner product space (V,(,)) with a not a scalar multiple of b. Define W-(a+ bt where t E R} which is a subset of V. Show that W is not a subspace of V. (b) Given u in V, define G(t) llabt-ull. Expand G(t) as a quadratic polynomial. (c) Suppose Projw+bt where t is some real nmber Thus G( 0. Using this fact and Part (b), give an explicit formula for t in terms of a, b, and u 4. Suppose you are skydiving on the fictional world of Erangel and your plane is traveling on the line y =一 + . You wish to land in the town of Pochinki which is located at the point u = (a) Find a, b e R2 such that W in Problem (3) is the line -[- "l (b) Find t using Problem (%) and use it to compute Prohru. Note the inner product (.) is the usual dot product on R2 (c) Note Projy u is the point on the line that minimizes its distance to u. Graph the line y- + u, and Projy u to verify this fact (d) How close will the plane ever get to Pochinki? 5. Iet V = C'[-1,11 be (quipped with the L2 innr product (f, g) = Li /(t)g(t) dt. (a) Projet tlu, alsolute value function f(t) = I, iftS() otto p21 1.11

Answer 1

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