## Subspace Spanned By Cosine and Sine Functions

## Problem 435

Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.

Define the map $f:\R^2 \to \calF[0, 2\pi]$ by

\[\left(\, f\left(\, \begin{bmatrix}

\alpha \\

\beta

\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.\]
We put

\[V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.\]

**(a)** Prove that the map $f$ is a linear transformation.

**(b)** Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.

**(c)** Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.

(This yields an isomorphism of $\R^2$ and $V$.)

**(d)** Define a map $g:V \to V$ by

\[g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.\]
Prove that the map $g$ is a linear transformation.

**(e)** Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.

(Kyoto University, Linear Algebra exam problem)

Add to solve later