Considej a 3 x 3 matrix A and a vector v in R3 such that A3v = 0, but A2v 0. a. Show that the vectors A2v, Av, v form a basis of R3. (Hint: It suffices to show linear independence. Consider a relation c\A2v + C2Av + civ = 0 and multiply by A2 to show that C3 = 0.) k Find the matrix of the transformation T(Jc) = Ax with respect to the basis A2v, Avy 0.

Week 2 Notes Calc II N Summation Notation; ∑ f (x) i=i Used for adding a lot of numbers or series of numbers with a pattern. Ex: 1+2+3+4+…+100 (easy to see pattern) Ex: .5+2+4.5+8+…+128 (not easy to see clear pattern) N ∑ f (x) i=i is a better way of writing these long patterned lists. 16i2 12 22 32 162 Ex: ∑ = + + +…+ =748 (this is the same i=12 2 2 2 2 pattern from above that didn’t seem to have a clear pattern) Properties of Summation Notation n n ai+¿∑ bi=∑ (i +i ) i=1 i=1