I'm revising for my linear exam and came across this question:
Consider the vector space P3 of all polynomials of degree <= 3, and
consider the subset S of P3 given by$$\text{ {S = a + b$x^2$ + c$x^3$ | a – b = 0} }$$
Show that S is a subspace of P3
So I started by checking the first axiom (closed under addition) to see if S is a subspace of P3:
Assume
polynomial 1 = $a_1 + b_1 x^2 + c_1 x^3$
polynomial 2 = $a_2 + b_2 x^2 + c_2 x^3$
p1+p2 = $a_1 + b_1 x^2 + c_1 x^3$ + $a_2 + b_2 x^2 + c_2 x^3$
= $ (a_1 + a_2) + (b_1 + b_2) x^2 + (c_1 + c_2) x^3$
But since a – b = 0, then a = b which means
= $ (a_1 + a_2) + (a_1 + a_2) x^2 + (c_1 + c_2) x^3$
= $ (a_1 + a_2) ( 1 + x^2) + (c_1 + c_2) x^3$
Which does not satisfy the first axiom.
But I was checking the correction, it turns out that it actually does satisfy the first axiom and S is a subspace of P3.
I still don't get why since there was no explanation in the paper, I'm hoping someone here could enlighten me and explain what I did wrong.
Thank you.
Best Answer
You already are on the right track.
$$\underbrace{(a_1+a_2)}_a-\underbrace{(a_1+a_2)}_b=0$$
Looking at the axiom carefully you will notice that $S$ is basically
$$ \{S = a (1+x^2) + cx^3|a,c\in\mathbb{R}\} $$