Generally speaking, one equation reduces the number of "free variables", and so the dimension, by 1. When you multiply a matrix, A, by the fixed vector, v, and set it equal to 0: Av= 0, each row of the matrix is one equation. Thus while the dimension of the space of all n by n matrices is the fact that Av= 0 gives n equations, reducing the dimension to .
P4 is the set of polynomials of degree 4 or less? Then your first space consists of all such polynomials that have only even powers: . The second is all such polynomials that have only odd powers: . It should be easy to find bases for those.