(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the subspace theorem to decide which of the following are real vector spaces with theusualoperations.

a) The set of all real polynomials of any degree.

b) The set of real polynomials of degree [tex]\leq n[/tex]

c) The set of real polynomails of degree exactly n.

2. Relevant equations

3. The attempt at a solution

I know how to do b) since the equation for the set of real polynomials of degree [tex]\leq n[/tex] is:

P_{n}= {a_{0}+ a_{1}x + a_{2}x^{2}+ ... + a_{n}x^{2}| a_{0}, a_{1}, .... , a_{n}[tex]\in[/tex] R }

And I can prove that it is closed under addition and scalar multiplication.

But I am not sure what the difference between the equation for b) (at most n) and a) (any n) and c) (= n) is.

Also I know b) is a real vector space but I would've thought that meant c) was too since b) includes degree = n. But the answers say a) is b) is but c isn't.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Linear Algebra Polynomial Vector Space

**Physics Forums | Science Articles, Homework Help, Discussion**