A subset of a vector space (the vector space of all polynomials) is a subspace (a vector space in its own right) if it is non-empty, is closed under addition, and is closed under scalar multiplication. This set is clearly non-empty.Question 1:
The set of all polynomials that take the value 1 at the point 0 is a vector space. (True or false)
Suppose f and g are two members of this set- they are polynomials such that f(0)= 1 and g(0)= 1. What about f+g? What is (f+g)(0)?
You might also think about 2f.
Because this is a subset of the set of all polynomials, you don't need to show things like "commutativity of addition" or a(f+ g)= af+ ag- those are true for all polynomials so true for this subset. As I said before you only need to prove this set is non-empty (obvious- f(x)= 0 is such a polynomial), that the set is "closed under additon" (if f(1)= 0 and g(1)= 0 then (f+g)(1)= 0) and "closed under scalar multiplication" (if f(1)= 0 then (af)(1)= 0 for any number a). Do you see the important difference between these two questions?Question 2:
The set of all polynomials that take the value 0 at the point 1 is a vector space. (True or false)
Yes, that is correct. Notice that above, I did not say anything about "contains the zero element"- you don't need to show that directly: if a set is not empty, say it contains f, and is closed under scalar multiplication, then it contains -f and it it is closed under addition, it must contain f+ (-f)= 0.Question 3:
The set of all polynomials of the form p(t)=a+bt+ct^2+t^3
I know that to determine if something is a vector space one would have to go through the list of axioms and see if any fail. For the third question, I suspect that it is not as it does contain the zero element since the coefficient of t^3 is 1<>0. As for the other questions I simply do not know or am not seeing it.
So it is true that any subspace must contain the "0" vector which, here, is the 0 polynomial. Since p(t) cannot be the 0 vector, this is not a subspace.
Pretty much any good linear algebra text will give examples of polynomial spaces.Lastly, I was wondering if someone has any suggestions as to a good textbook that may elaborate on the topic of polynomials and vector spaces.