# determinant of linear maps

• November 26th 2011, 06:36 AM
beano2913
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
• November 26th 2011, 10:37 AM
FernandoRevilla
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.

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