determinant of linear maps

Let V be the vector space of functions which are linear combinations of

1, cos x, sin x, cos 2x, sin 2x, . . . , cos(N x), sin(N x).

Let φ : V → V be the linear map

φ(f ) =∫ f (u) du. between x and 0

Calculate det φ.

I am not sure what basis to use for V when solving this. Any help would be appreciated

Re: determinant of linear maps

Quote:

Originally Posted by

**beano2913** Let V be the vector space of functions which are linear combinations of 1, cos x, sin x, cos 2x, sin 2x, . . . , cos(N x), sin(N x). Let φ : V → V be the linear map φ(f ) =∫ f (u) du. between x and 0 Calculate det φ.

$\displaystyle \varphi$ is not well defined. Consider for example $\displaystyle n=1$ then, $\displaystyle V=\mathcal{L}\{1,\cos x,\sin x\}$ and $\displaystyle \varphi (1)=\int_x^01\;du=-x\not \in V$ .