Thread: Three quick questions - Vector spaces and polynomials

Good day to all,

I have read the section on general vector spaces and there was a passing by mention of polynomials as a vector space. Furthermore, my teacher has mentioned nothing about polynomials in relation to vector spaces. As I was doing the end of chapter exercises, I came across two questions that I do not understand:

Question 1:

The set of all polynomials that take the value 1 at the point 0 is a vector space. (True or false)

Question 2:

The set of all polynomials that take the value 0 at the point 1 is a vector space. (True or false)

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.

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.
Kindest regards

Originally Posted by gate13

Good day to all,

I have read the section on general vector spaces and there was a passing by mention of polynomials as a vector space. Furthermore, my teacher has mentioned nothing about polynomials in relation to vector spaces. As I was doing the end of chapter exercises, I came across two questions that I do not understand:

Yes, you can add two polynomials and you can multiply a polynomial by a real number and they satisfy all "axioms" for a vector space.

Question 1:

The set of all polynomials that take the value 1 at the point 0 is a vector space. (True or false)

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.

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.

Question 2:

The set of all polynomials that take the value 0 at the point 1 is a vector space. (True or false)

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 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.

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.
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.

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.
Kindest regards

Pretty much any good linear algebra text will give examples of polynomial spaces.