Quick question about notation (Linear Algebra)

[calstudent]

$$p\in P_3(\mathbb{F})$$

What does $$\overline{p(z)}$$ mean?

I would guess that it's related to the complex conjugate, but I'm not sure. For context, I'm dealing with an inner product space defined by $$\langle p,q\rangle=\intop_{0}^{1}p(z)\overline{q(z)}dz$$

Thanks!

 

[Dick]

Yes, it's a complex conjugate.

 

[HallsofIvy]

that's an "inner product" of complex valued functions. The reason for the complex conjugate is so that
$$<f(z),f(z)>= \int_0^1 f(z)\overline{f(z)}dz$$
will be a non-negative real number.

 

[calstudent]

Inner Product Space Norm

Thanks for the help so far. I'm trying to find the norm for the this inner product. So far I have:

$$\Vert p\Vert=\sqrt{\left\langle p,p\right\rangle }=\sqrt{\intop_{0}^{1}p(z)\overline{p(z)}dz}=\sqrt{\intop_{0}^{1}|p(z)|^{2}dz}$$

I also know that $$p(z)$$ can be denoted $$\sum_{i=0}^{n}a_{i}z^{i}$$ but I don't know how to connect the two beyond placing the sum within the absolute value.

Is there a way to simplify $$|\sum_{i=0}^{n}a_{i}z^{i}|^{2}$$?

 

[Dick]

You've already written the norm of the inner product. Do you want to write it in terms of the a_i's? Is P_3 third degree polynomials? Or second? Either way it's going to be kind of uselessly complicated. If it's second, e.g. write (a0+a1*z+a2*z^2) times the complex conjugate and integrate over the real variable z in [0,1].