Showing a basis for a polynomial I am having trouble with this basic basis problem. Need to show that $\{z^4,z^4-z^3,z^4-z^3+z^2,z^4-z^3+z^2-z,z^3-1\}$ is basis for $P_4$. I figured out that it is linearly independent but having problem show that it spans $P_4$

From $z^4$ and $z^4-z^3$, you get $z^3$ by a linear combination.
From $z^4-z^3+z^2$, you get $z^2$ since you have $z^4$ and $z^3$.
From $z^4-z^3+z^2-z$, you get $z$ since you have $z^4$, $z^3$, $z^2$.
From $z^3-1$, you get $1$ since you have $z^3$.
So, $z^4$, $z^3$, $z^2$, $z$, $1$ are all in the subspace generated by $z^4$, $z^4-z^3$, $z^4-z^3+z^2$, $z^4-z^3+z^2-z$, $z^3-1$. Therefore, this subspace must be the whole space $P_4$.

More systematically, write the given polynomials with respect to the basis $z^4$, $z^3$, $z^2$, $z$, $1$; you'll get a triangular matrix that is invertible because it has $\pm1$ in the diagonal.
