Question

W is the set of all polynomials in P_2 such that p(1)=0 is a subspace of P_2. Find a basis for W...How do I find this? I know that a particular set is a basis for a given vector space if the set is linearly independent and spans that vector space, but I don't know how to figure out that set for this given space.

Answer 1

A basis for any polynomial of degree n or less is {1,t,t^2,...,t^n}, which is called the natural or standard basis of a polynomial. Use that to find a basis for your P_2 polynomial.

Answer 2

My heuristic guess is (t -1, t^2 -1).

We know that we need linear combinations of 1, t and t^2 that are linearly independent. In general, a restriction like the above takes away one degree of freedom so the dimension of our space is 2.

Answer 3

I found the answer for the problem and you're correct davismj, but I don't quite understand how you came to your conclusion. How does the problem take away one degree of freedom? I donno, I'm still a bit confused. ha. Why does it have to be a linear combination of 1, t, and t^2? Wouldn't p(1)=1^2+1+1=3 and not 0? It seems like the space would be something like t^2, t, -2 ha..then p(1)=1^2+1-2=0..but even if that was correct, I still would not know how to find a basis for it ha.

Answer 4

The answer works perfectly! For example, 5(t-1)+2(t^2-1)= 2t^2+5t-7.....p(1)=2(1)+5(1)-7....p(1)=0....any coordinates chosen works for the basis {t-1,t^2-1}. I understand why this problem is correct; I just don't understand how you discovered it. ha...I want to be able to solve similar problems in the future. Can you explain exactly what went through your mind while figuring it out? ha