I need a math genius to help me with this!!! (vector spaces)

I was reading my linear algebra textbook, and there is something that confused me...so hopefully someone will try to clarify it...

In the textbook, it says "For n>= 0 the set P_n of polynomials of degree at most n consists of all polynomials of the form p(t)=a_0 + (a_1)(t) + (a_2)(t^2) + ... + (a_n)(t^n) where the coefficients a_0...a_n and the variable t are real numbers. The degree of p is the highest power of t whose coefficient is not zero. If p(t)= a_0, the degree of p is zero. If all the coefficients are zero, p is called the zero polynomial. The zero polynomial is included in P_n, even though its degree, for technical reasons, is not defined.

MY QUESTION:

So here it explains how P_n is a vector space. I did not type the part where it talks about how (p+q)(t) = p(t) + q(t) and how (cp)(t) = cp(t)...etc, because I understood them. The part that I did not understand is how the zero polynomial is included in P_n. As I typed above, the textbook says "The zero polynomial is included in P_n even thout its degree, for ntechnical reasons, is not defined." What confuses me is that...If we substituted t=0 in the equation p(t)=a_0 + (a_1)(t) + (a_2)(t^2) + ... + (a_n)(t^n), then p(0) = a_0...we cannot necessarily say that a_0 = 0, because a_0 can be any real number ("the coefficients a_0...a_n ...are real numbers"). So how is the zero polynomial included? Shouled p(0)=0 for the zero polynomial to be included? Can anyone please explain this to me, because I find it very confusing.

Thank you in advance...

Hello Artus,

I think you misinterpret the point of the paragraph. a_0 can be any number, which includes 0. We only need p(x)=0 to find that 0 is included in the set of all polynomials. This is achieved by setting all the coefficients, a_0 to a_n, to 0.

i.e.: a_0=0, a_1=0, ... , a_n=0

We need not select a value of t here as p(x)=0 for all values of t if and only if all the coefficients are set to zero. This is why we can p(x)=0 in the set of all polynomials.

I hope this makes a tad more sense.

TLDR: Your coefficients are a_0 to a_n not the respective powers of t.

Darren

Oh...ok I understand it now...thank you...I don't know why I was thinking about it differently. I guess I didn't read "all coefficients are zero" carefully; I thought t had to be zero, for some reason...anyways, thanks.