(1) Let by(t) = 4+2+, bz(t) = -6 +8t, (t) = 1 +t, cz(t) = 1 - t, and let B = {61(t), 62(t)} and C = {ci(t), cz(t)}. (a)

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1 Let By T 4 2 Bz T 6 8t T 1 T Cz T 1 T And Let B 61 T 62 T And C Ci T Cz T A 1 (90.92 KiB) Viewed 24 times

(1) Let by(t) = 4+2+, bz(t) = -6 +8t, (t) = 1 +t, cz(t) = 1 - t, and let B = {61(t), 62(t)} and C = {ci(t), cz(t)}. (a) Show that B and C are bases for P. (b) Let p(t) = 361(t) + 2b2(t). What is {p(t),B? (e) Since C is also a basis for Ps, there is also a coordinate vector for p(t) with respect to C, and it is reasonable to ask how (p(t) c is related to (P(t)]3. Recall that a coordinate transformation respects linear combinations - that is [rx + sy s = r[x]s + s[y]s for any vectors x and y in a vector space with basis S, and any scalars r and s. Use the fact that p(t) = 361(t) + 2b2(t) and the linearity of the coordinate transforma- tion with respect to the basis C to express (p(t) c in terms of [bi(t) c and (62(t)]c (don't actually calculate (bu(t)c and (b2(t)\c yet, just leave your result in terms of the symbols (61(t)lc and (62(t).c.) (d) The result of part (c) can be expressed as a matrix-vector product of the form [p(t)}c = P(p(t)]s. Describe how the columns of the matrix P are related to [bi(t)}e and (b2(t)}c. (e) Now calculate (bz(t) c. (62(t)]c), and (p(t)]c. Determine the entries of the matrix P and verify in this example that (p(t)}c = P[p()]B. (2) The matrix P that we constructed in problem (1) allows us to quickly and easily switch from coordinates with respect to a basis B to coordinates with respect to another basis C. providing a way to effectively transition from one coordinate system to another as described in the introduction. This matrix Pis called a change of basis matrix. In problem (1) we explained why the change of basis matrix exists, and in this problem we will see another perspective from which to view this matrix. Let B = {bı, b2} and C = {ci, c2} be two bases for a vector = space V. The change of basis matrix P from B to C has the property that P[x]B = xc for every vector x in V. We can determine the entries of P by applying this formula to specific vectors in V. (a) What are (bu B and (b2]B? Why? (b) If A is an n x n matrix and e1, 62, ..., er are the standard unit vectors in R” (that is, e; is the ith column of the n x n identity matrix), then what does the product Ae; tell us about the matrix A? (c) Combine the results of parts (a) and (b) and the equation P(x = (x c to explain why P = (bilc b2]c).