Find matrix representation with respect to the basis

[Kronos1]

The set of all solutions of the differential equation

$$\d{^2{y}}{{x}^2}+y=0$$

is a real vector space

$$V=\left\{f:R\to R \mid f^{\prime\prime}+f=0\right\}$$

show that $$\left\{{e}_{1},{e}_{2}\right\}$$ is a basis for $V$, where

$${e}_{1}:R \to R, \space x \to \sin(x)$$

$${e}_{2}:R \to R, \space x \to \sin\left(x+\frac{\pi}{4}\right)$$

Show that

$$D:V \to V, \space y \to \d{x}{y}$$

is a linear transformation and find it's matrix representation with respect to the basis above

 

[Ackbach]

What have you done so far? Also, what theorems are you allowed to invoke? Lastly, are you sure it's not
$$D:V\to V, y \to \frac{dy}{dx}?$$

 

[Kronos1]

you are correct it is $$D:V \to V, \space y \to \d{y}{x}$$

so far I have set $${e}_{1}\space and\space {e}_{2}=V$$ to prove that the definition of $$f^{\prime\prime}+f=0$$ which was true for both of them. Now to my knowledge to prove a basis I need to prove that they are linearly independent which I think can be done by showing that $${c}_{1}{e}_{1}+{c}_{2}{e}_{2}=0 \space for \space c\in R$$ which as they both equal 0 is true for all $c$.

Assuming the above part is right I don't know how to go about the next part?

 

[Deveno]

you are correct it is $$D:V \to V, \space y \to \d{y}{x}$$

so far I have set $${e}_{1}\space and\space {e}_{2}=V$$ to prove that the definition of $$f^{\prime\prime}+f=0$$ which was true for both of them. Now to my knowledge to prove a basis I need to prove that they are linearly independent which I think can be done by showing that $${c}_{1}{e}_{1}+{c}_{2}{e}_{2}=0 \space for \space c\in R$$ which as they both equal 0 is true for all $c$.

Assuming the above part is right I don't know how to go about the next part?

That is not what linearly independent means. It means that if: $c_1,c_2$ are real constants such that:

$c_1e_1(x) + c_2e_2(x) = 0$, for ALL $x \in \Bbb R$ that we MUST HAVE $c_1 = c_2 = 0$.

Since both $\sin(x)$ and $\sin(x + \frac{\pi}{4})$ are periodic (with period $2\pi$) it suffices to consider $x \in [0,2\pi)$.

Suppose $x = \dfrac{\pi}{2}$. What does that tell you about what $c_2$ must be in terms of $c_1$?

Next, suppose $x = 0$. What does that tell you about what $c_2$ must be?

Proving spanning is even harder: you must show that ANY solution to:

$f'' + f = 0$

can be written in the form $c_1e_1 + c_2e_2$ for some real numbers $c_1,c_2$.

This is not a trivial problem. To see what I mean, note that:

$f(x) = \cos(x)$ is an element of your vector space.

 

[Ackbach]

You can also use the Wronskian to show linear independence.