Find the basis when integration is in the condition

Question

Let $V$ be the set of all polynomial $f(x)$ in $P_2$ s.t. $\int_{0}^3 f(x) dx =3f(1) $

If $V$ is a subspace of $P_2$ find a basis of $V$.

Can somebody help me get started? The integral condition kind of scares me.

Answer 1

There is nothing to be scared about with the integral, as long as you can evaluate basic integrals, which I'm sure you can.

Any polynomial in $P_2$ has the form $f(x)=a+bx+cx^2$. This satisfies your condition if $$\int_0^3 a+bx+cx^2\,dx=3(a+b+c)\ .$$ If you calculate the integral and simplify this comes down to $$b=-4c\ .$$ Therefore $$\eqalign{V &=\{\,a+bx+cx^2\in P_2\mid b=-4c\,\}\cr &=\{\,a-4cx+cx^2\mid a,c\in{\Bbb R}\,\}\cr &=\{\,a(1)+c(-4x+x^2)\mid a,c\in{\Bbb R}\,\}\cr &=\mathop{\rm span}\{\,1,\,-4x+x^2\,\}\ .\cr}$$ This shows that $\{\,1,\,-4x+x^2\,\}$ is a spanning set for $V$; it is easy to show that it is linearly independent; so it is a basis for $V$.