Linear Algebra - Dimension of Subspace How do I go about finding the dimension of the subspace:
$$S:= \{{ p(x) \in P_4: p(-x)= -p(x)\space \forall x \in \mathbb{R} \}}\space of \space P_4$$
My textbook says $\dim(P_n)=n+1$, but this does not give me the correct answer (the correct answer is 2). All help is appreciated. 

So the set $S$ contains the polynomials $\in P_4$ such that $p(-x) = -p(x)$. So lets start by determining the form of these polynomials. Notice how for a monomial with an  even degree e.g. $x^2$ would return the same value if you were to put $-x$ or $x$ however a monomial with an odd degree e.g. $x^3$ would return minus the value you would get if you  put $-x$ instead of $x$. So the only odd monomials that belong to $P_4$ are $x,x^3$ these then would form your basis as they satisfy the property $p(-x) = -p(x)$ and are linearly independent. Therefore the dimension of the $S$ is $2$.
