# Kernel of a polynomial

Let's say we have a 2nd degree polynomial $a+bx+cx^2$ and it is given that $T:P2\rightarrow R$ given by $T(p)=\int_{0}^{1}p(x)dx$

We are asked to find the kernel of $T$. Now, I know that depending on which constant I end up solving for after integration I will get that the kernel is the span of 2 different vectors , but once again , it depends on which constant I solved for. Does it matter whether I solve for c ( the coefficient of $x^2$) or a and then do the substitution?

I guess a summary my question is.. in $\text{Ker}(T)= \text{span}(v_1,v_2)$, are $v_1$ and $v_2$ unique (the only solution) ? or could they be different but still the right answer for Ker(T)?

• Hmm. What's P2? – MPW May 15 '16 at 2:36
• 2nd degree poly – Peter B. May 15 '16 at 2:36
• You mean the set of all polynomials of degree 2? – MPW May 15 '16 at 2:37
• Yes that's exactly what I mean – Peter B. May 15 '16 at 2:38

The kernel of a transformation is a subspace. When you say the kernel is the span of two vectors $v_1$ and $v_2$ you are saying that these vectors form a basis. There are an infinite number of choices you can make for the basis vectors. As long as they are independent and span your subspace, they are a valid choice