u/sincethelasttime

An exam question I faced asked if U was a valid subspace if U was the set of polynomials p(x) such that p(5) = 5, x is real. My proof used the fact that a subspace must contain a vector -v such that v + -v = 0v. Let g(x) = v and h(x) = -v.

Then g(x) + h(x) = 0v. Set x = 5

g(5) + h(5) = 0v.

5 + 5 = 0v.

10 = 0v

Is this valid?

My concerns are -

1.) Do subspaces have to contain a -v for every v to be valid?
2.) Is adding 5 +5 = 10 a valid operation if we haven't defined scalar+scalar addition, only that fields are closed under scalar addition in this way.

I am aware I should have used the 0(x) property in this proof