# Length of a polynomial vector?

1. Dec 12, 2013

### PhizKid

1. The problem statement, all variables and given/known data
S = {1, x, x^2}

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

2. Relevant equations
$\sqrt{v \cdot v}$

3. The attempt at a solution
I don't know the components of each vector, so how can I perform the dot product?

2. Dec 12, 2013

### Office_Shredder

Staff Emeritus
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?

3. Dec 12, 2013

### 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?

4. Dec 12, 2013

### Office_Shredder

Staff Emeritus
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.

5. Dec 12, 2013

### PhizKid

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

6. Dec 12, 2013

### Staff: Mentor

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.