# Matrix representation for the endomorphism $f \mapsto f''$

I'm asked to find the matrix that represents the endomorphism $g$ with respect to the basis that I found. I'm told that the endomorphism is defined by

$$\Delta : S_3 \to S_3, \ \ \ \Delta (f)(x)=f''(x)$$

and $S_3$ is the vector space spanned by the basis $$\left\{ \frac{1}{\sqrt 2} , \cos x, \cos 2x, \cos 3x\right\}$$ How should I approach this exactly? I know the orthogonal basis but nothing else. Thanks!

• Once you've answered the above two questions, the columns of the matrix representation are the images of the basis (column) vectors by the linear map. – Arthur Jan 8 '18 at 8:56
• That's not a basis for the vector space of all functions from $\Bbb R$ to $\Bbb R$. It's way too small. – Arthur Jan 8 '18 at 8:59
• @Arthur Yes sorry, I have $S_{N}$ that is generated by the basis $(1, \cos (x), \cos (2x), \cos(3x), ..., \cos(Nx))$ but we're only looking at the subspace $S_{3}$ – Niktaneous Jan 8 '18 at 9:04

• $\Delta\left(\frac1{\sqrt2}\right)=0$;
• $\Delta\bigl(\cos(x)\bigr)=-\cos(x)$;
• $\Delta\bigl(\cos(2x)\bigr)=-4\cos(2x)$;
• $\Delta\bigl(\cos(3x)\bigr)=-9\cos(3x)$,
the matrix that you are interested in is $\begin{bmatrix}0&0&0&0\\0&-1&0&0\\0&0&-4&0\\0&0&0&-9\end{bmatrix}$.