Find dimension of even polynomials Let $V$ be a the vector space over $\mathbb R$ of all polynomials with real coefficients. Let $W$ be the subset of all polynomials with only even powers in their expression. 
So $p(X) \in W$ means $p(x)=\sum_{n=0}^k a_nx^{2n}$.
I showed that $W$ is a subspace of $V$, but I need to find out the dimension of $W$. Would I need to find a basis for $W$ first? How should I approach this?
Thank you. 

Your space $W$ is infinite dimensional. It is easy to see that every polynomial in  $W$ can be expressed as a (finite) linear combination of the polynomials $1, x^2, x^4, x^6, x^8,\dots$.
For completeness, one should show that these polynomials form a linearly independent set, and are therefore a basis for $W$.  This follows immediately from the definition of equality for polynomials. For an alternative argument, suppose that $P(x)=a_0+a_1x^2+a_2x^4+\cdots +a_nx^{2n}$ is identically $0$. We show that all the $a_i$ are $0$. 
If the $a_i$ are not all $0$, we may assume without loss of generality that $a_n\ne 0$. From
$$0\equiv a_0+a_1x^2+a_2x^4+\cdots +a_nx^{2n},$$
by differentiating $2n$ times we get
$$0\equiv (a_n)(2n)!,$$
which is false.
