here I am, studying for my linear algebra midterm when I stumble upon the following problem:

Code:

Consider the vectors cos(x+y) and sinx in C[-pi, pi]. for what values of y will
the two vectors be linearly dependent?

well the linear part is quite easy so I will do that here. But what gets me is the trig afterwards:

Code:

W[cos(x+y), sinx] = | cos(x+y) sinx |
|-sin(x+y) cosx |
= cos(x+y)cosx + sin(x+y)sinx = 0

I wont get into the symantics of linear algebra too much, but basically for the vectrs to be linearly dependent that determinant has to equal zero. Now the problem is for what values of y will this equation hold:

Code:

cos(x+y)cosx + sin(x+y)sinx = 0

anyone?