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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
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