Basis of the polynomials

Question

How can I show that the polynomials: $(1, 1+x, x+x^2)$ are the basis of $P_2$, the vector-space of polynomials of degree $2$?

I tried to show that they are linear dependent thus, $$a + b(1+x) + c(x+x^2) = 0$$ However, I fail to continue.

Answer 1

$$a + b(1+x) + c(x+x^2) = 0$$ must hold for all $x$.

For $x=-1,0,1$, this reduces to $$\begin{cases}a=0,\\a+b=0,\\a+2b+2c=0.\end{cases}$$

The conclusion is clear.

Answer 2

For another approach, write $$ \pmatrix{ 1 \\ 1+x \\ x+x^2} = \pmatrix{ 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1} \pmatrix{ 1 \\ x \\ x^2} $$ and note that the matrix is invertible.

More generally, this argument proves that a set containing exactly one polynomial of each degree is a basis for the space of polynomials up to a certain degree. The corresponding matrix will be lower triangular with no zero diagonal entries.