Understanding how to check subspaces

Question

So here's my problem. I'm trying to do some practice on Subspace of vectors, and this was a problem in the book.

Determine if the given set is a subspace of $P_n$, the set of polynomials of degree at most $n$, for an appropriate value of $n$. Justify your answers.

1. All polynomials of the form $p(t)=a+t^2$, where a is in $R$.
2. All polynomials in $P_n$ such that $p(0)=0$.

For one thing, I don't understand how to check Subspaces. I know the following for them:

A subspace of a vector space $V$ is a subset $H$ of $V$ that has three properties:
a. The zero vector of $V$ is in $H$?
b. $H$ is closed under vector addition. That is, for each u and v in $H$, the sum u + v is in $H$.
c. $H$ is closed under multiplication by scalars. That is, for each u in $H$ and each scalar $c$, the vector $c$u is in $H$

How do I use this information to check the problem above? What's the actual math involved for this? My book only gives me this info and doesn't really explain anything about it.

Answer 1

1. It is not: the zero polynomial is not in the set.

2. Let us check each property:

a. The zero vector is in the set as if $p(t)=0 \Rightarrow p(0)=0$. Checked.

b. Let us denote by $p_1(t)$ and $p_2(t)$ two polynomials which satisfy $p_1(0)=0=p_2(0)$. Then $(p_1 + p_2)(0)=p_1(0)+p_2(0)=0$. Checked.

c. If $p(0)=0$, then $(cp)(0)=cp(0)=0$. Checked.

Then the set of all polynomials that satisfy $p(0)=0$ is a subspace.

Answer 2

Quick elaboration on the criteria you have above: a subspace $H$ of a vector space $V$ is a subset that is itself a vector space under the vector addition and scalar multiplication of $V$.

It can be shown that the conditions you listed are necessary and sufficient for a subset $H$ to be a subspace of $V$. Thus, the properties you have above constitute a characterization of a subspace and not a definition.