MTH 154 Lecture 2: Section 6.3

If Tj : P(R) rightarrow P(R) be the linear operator sending a polynomial to its j-th derivative. Prove that T1,... ,Tn are linearly independent in L(P(R)). Let T : V rightarrow W be a linear transformation between F-vector space V and W of equal dimension. Prove that there exist basis beta of V and gamma of W such that [T]gamma beta is a diagonal matrix whose entries are 1 or 0.

Evaluate the work done between point 1 and point 2 for the conservative field F. F = 5 sin 5x cos 6y cos 2zi + 6 cos 5x sin 6y cos 2z j + 2 cos 5x cos 6y sin 2zk ; P1 (0, 0, 0), P2 W = 0 W = -2 W = 2 VV = 1

Let u = 2i + j, v = i + j, and w = i - j. Find scalars a and b such that u= av + bw.