Mathematics 3C, Fall 2011 Selected Solutions to Worksheet 4, TA Grace Kennedy Summary: Mathematics 3C, Fall 2011 Selected Solutions to Worksheet 4, TA Grace Kennedy NAME: MY WEBSITE: http://math.ucsb.edu/kgracekennedy/Fall2011 3C.html Course Website: Access through GauchoSpace. 4. Is the following set of functions in C[0, 1] linearly independent? S = {cos(t), cos(2t), cos(3t)} Intuitively, when you try to apply the definition of linear independence: do there exist nontrivial a, b, c R so that a cos(t) + b cos(2t) + c cos(3t) = 0, you should think `it would be weird if you could scale trig functions so that they cancel out with scalars on the inside...' Scaling trig functions on the outside changes the altitude while scaling them on the inside changes their period, or how close together the humps are, but not the height. The rigourous way to show this is by using the Wronskian, which you learned in lecture on Thursday the 27th, but this problem would be tedious with that method. I intended this to be approached intuitively. 5. Is the following set of functions in C[0, 1] linearly independent? S = cos( 2 Collections: Mathematics