1. Kernel and Image

The following at linear transformations. Find the kernel and images:

1) φ : P4 ->R given by φ(p(x)) = p′′(0), where p′′(x) denotes the second derivative

2) 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

2. For (1), remember that a polynomial evaluated at 0 is simply the constant term of the polynomial. So, polynomials whose second derivative has a constant term of 0 would be in the kernel of that map. What kinds of polynomials are these? As for the image, is there any number that could not possibly be the constant term of the second derivative of some polynomial?

And for (2)... I'm not quite sure what $\displaystyle \mathbb{R}!\mathbb{R}$ means...

3. Its suppose to be R->R not R!R, same with V->V. Sorry about that

4. Originally Posted by skittle
Its suppose to be R->R not R!R, same with V->V. Sorry about that.
So I guess you mean that $\displaystyle V=\{\alpha \cos(x)+\beta \sin(x)|\alpha,\beta\in\mathbb{R}\}$? Just making sure I completely understand the question before I lead you in the wrong direction.

5. Thats what I got from the question

6. Okay.

So if $\displaystyle f(x)=\alpha \cos(x)+\beta \sin(x)$ is such a function, then what does $\displaystyle f^{\prime}(x)+f(x)$ look like? When could it possibly be 0? And can every element of $\displaystyle V$ be written in this form, or are there some elements of $\displaystyle V$ that don't look like this?