1. ## Injective, Surjective

Is the following linear transformation injective? surjective?

φ : V ->V given by φ(f) = f+ f, where V is the subspace of the space of smooth functions R ->R spanned by sin and cos, and fdenotes the derivative

I know what you need inorder to say whether or not its injective or surjective, but Im confused on the steps to find out if it is. If someone could show me the steps that would be alot of help.

Thanks

2. Hint

The matrix of $\displaystyle \varphi$ with respect to the basis $\displaystyle B=\{\sin x,\cos x\}$ of $\displaystyle V$ is

$\displaystyle A=\begin{bmatrix}{1}&{-1}\\{1}&{\;\;\;1}\end{bmatrix}$

Now, find

$\displaystyle \dim(\ker f),\;\dim(\textrm{Im}f)$ .

Fernando Revilla