Vector spaces and basis I came across this problem on vector space basis:
Verify that $2-x^2, x^3-x, 2-3x^2$ and $3-x^3$ form a basis for $P^4$ and express each of the polynomial $x^2$ as a linear combination of this basis.
I have tried solving the problem with my knowledge of vector space but still don't know how to go about it.
Any helpful solution to this?

1) Let's verify that the system of vectors $(2−x^2,x^3−x,2−3x^2, 3−x^3)$(1) is a basis for $P^4$. Write the canonical basis for $P^4$: $(1 ,x ,x^2 ,x^3)$(2) and decompose vectors of the system (1) by basis (2):
\begin{pmatrix}
2 & 0 & 2 & 3\\
0 & -1 & 0 & 0\\
-1 & 0 & -3 & 0\\
0 & 1 & 0 & -1\\
\end{pmatrix} 
Now we'll proof linear independence of (1):
$
A=\begin{pmatrix}
2 & 0 & 2 & 3\\
0 & -1 & 0 & 0\\
-1 & 0 & -3 & 0\\
0 & 1 & 0 & -1\\
\end{pmatrix}
\sim
\begin{pmatrix}
1 & 0 & -1 & 3\\
0 & -1 & 0 & 0\\
-1 & 0 & -3 & 0\\
0 & 0 & 0 & -1\\
\end{pmatrix} 
\sim
\begin{pmatrix}
1 & 0 & -1 & 3\\
0 & -1 & 0 & 0\\
0 & 0 & -4 & 3\\
0 & 0 & 0 & -1\\
\end{pmatrix}
\sim
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
\end{pmatrix}$
Proofed,(1) is linear independent, $rankA=rank(1)=4$. But it is maximal linear independent for $P^4$( if we add any vector to (1), it becomes linear dependent (definition of max. linear independent system)($dimP^4=rank(1)=4$)), so we can state that (1) is a basis for $P^4$.
2)Since (1) is a basis for $P^4$, we can decompose any vectors of the $P^4$ by (1)(definition of basis), including each of the polynomial $x^2$. 
