In the vector space $P_3$ of all $p(x)=a_0+a_1x+a_2x^2+a_3x^3$, let $S$ be the subset of polynomials with $\int_0^1p(x)dx=0$. Verify that $S$ is a subspace and find a basis.
I'm not sure where to start on this question. Can somebody help please?
[Math] Verifying that the subset of polynomials with vanishing integral is a subspace
integrationlinear algebrapolynomials
Best Answer
Hint $\ $ The map $\rm\: L(f) = \int_0^1 f\:$ is an $\mathbb R$-linear map between the $\mathbb R$-vector spaces $\rm\:P_3$ and $\mathbb R$, hence its kernel $\rm\: K = ker\ L = \{f : Lf = 0\}$ forms a subspace of $\rm\:P_3$. If you haven't yet learned this general fact then you can prove it now with no greater effort than that required for your special case, but with the benefit that you now have a more general result which yields greater conceptual insight.
To find a basis of $\rm\:K,\:$ note that for $\rm\:n\in \{1,2,3\},$ $\rm\: 0 \ne x^n - L(x^n)\in K,\:$ by $\rm\:L(r) = r\:$ for $\rm\:r\in \mathbb R$. But $\rm\:dim\ K \le 3,\:$ (why?) and these $3\:$ elements of $\rm\:K\:$ are independent, hence $\ldots$