Given W with a basis, how do I find the basis for orthogonal complement of W?

Question

Let $V$ be the inner product space of $P_{3}$ and $W$ with basis $\{1, \mathbb t^2 \}$. Find a basis for $W^\perp$.

I'm not quite sure what I need to do here first. Do I need to use Gram-Schmidt? What are the steps that I need to take?

Answer 1

It does depend on how the inner product on the space is defined. I am not sure myself what is considered as "standard" in this case. But for example, let's say the inner product is defined as $$\langle p_1(t),p_2(t) \rangle = \int_{-1}^1 p_1(x)p_2(x) dx.$$ (you can verify that this is an inner product on $P_3$)

In this case you don't even need to use Gram-Schmidt. Why? Because the integral of an odd power will yield an even power, which will be zero with the limits of integration as defined above - this implies that using the remaining two vectors in the standard basis of $P_3$ you will have $\langle 1,t \rangle = \langle t^2,t \rangle =0$ and $\langle 1,t^3 \rangle = \langle t^2,t^3 \rangle =0$. (You can verify that this is indeed true by doing the integration)

So then, with the inner product as defined above $\{t,t^3\}$ is a basis for $W^\perp$.