Linear Algebra: Polynomials Basis Consider the polynomials $$p_1(x) = 1 - x^2,\;p_2(x) = x(1-x),\;p_3(x) = x(1+x)$$
Show that $\{p_1(x),\,p_2(x),\,p_3(x)\}$ is a basis for $\Bbb P^2$.
My question is how do you even go about proving that these polynomials are even independent? Are there certain rules I should know?

No particular rule for polynomials: they are elements of a vector space of dimension $3$.
Since $\{1;x;x^2\}$ is obviously a basis for your space, you can simply show that the matrix
$$
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
-1 & -1 & 1
\end{bmatrix}
$$
has rank $3$, which is done by a simple elimination.
Why is this true? Because the columns of this matrix are the coordinates of $p_1$, $p_2$ and $p_3$ with respect to the basis $\{1;x;x^2\}$ and a set of vectors is linearly independent if and only if the set of their coordinate vectors is linearly independent.
