Length of a polynomial vector?

[PhizKid]

Homework Statement

S = {1, x, x^2}

Find ||1||, ||x||, and ||x^2||.

Homework Equations

$\sqrt{v \cdot v}$

The Attempt at a Solution

I don't know the components of each vector, so how can I perform the dot product?

 

[Office_Shredder]

You have been told at some point what the inner product on the space of functions is (it probably involves an integral). Can you tell us what it is?

 

[PhizKid]

$<f,g> = \int_{0}^{1} fg \textrm{ } dx$ is what was given previously. I didn't think it was relevant to find the norm but I guess it is somehow?

 

[Office_Shredder]

When you write that the length of a vector is
$$\sqrt{v \cdot v }$$
what you are really writing is
$$\sqrt{ \left<v,v \right> }$$

In any inner product space you can define the length of a vector in this way, even if the inner product is not actually a dot product.

 

[PhizKid]

Ah, okay. So the respective lengths are just $\sqrt{1}$, $\sqrt{\frac{1}{3}}$, and $\sqrt{\frac{1}{5}}$?

 

[Mark44]

Ah, okay. So the respective lengths are just $\sqrt{1}$, $\sqrt{\frac{1}{3}}$, and $\sqrt{\frac{1}{5}}$?

Since you ended with a question mark, you're not sure. Please show us what you did to get these, rather than making us do that work.