Inner Product Spaces

Question

My text as the following problem:

Let $V$ be the Euclidean space $P_3$ with the inner product defined in Example 2. Let $W$ be the subspace of $P_3$ spanned by $[t-1,t^2]$ Find a basis for $W^{\perp}$.

Inner product defined in Example 2: Let $V = P$; if $p(t)$ and $q(t)$ are polynomials in $P$, we define $\int_0^1 p(t)q(t)\,\mathrm{d}t$ as the inner product.

I am having trouble trying to figure out what I am supposed to do with this information. I assume $p(t) = t-1$ and $q(t) = t^2$ ? Then I can find the inner product by just evaluating it.

Answer 1

You look for a third polynomial which is orthogonal to both $\{(t-1),t^2\}$, within the space of polynomials of degree 3 - which has dimension 4. A standard base for $P_3$ would be $\{1,t,t^2,t^3\}$. Now the task to be done is:

• find two polynomials linear independent of $\{(t-1),t^2\}$
• and then apply the Gram-Schmidt orthonormalisation process

that should solve the problem (hopefully)