[Math] Show that the following set is a basis of P2

Question

Show that $\{a+bx+ cx^2, a_1+ b_1 x + c_1 x^2, a_2 + b_2 x + c_2 x^2\}$ is a basis of $P_2$ if and only if $\{(a,b,c) , (a_1, b_1, c_1), (a_2, b_2, c_2)\}$ is a basis of $\mathbb{R}^3$.

I was thinking about using the augmented matrix but I don't know how to start.

Answer 1

If $v_1,\dots,v_n$ are vectors in $\mathbb{R}^n$, then they form a basis precisely when the matrix $$\begin{pmatrix} | & | & | \\ v_1 & \dots & v_n \\ | & | & | \end{pmatrix}$$ has non-zero determinant. To be clear, the columns of the matrix are the vectors $v_i$.

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Note that if you express the vectors in the first collection with respect to the basis $\{1,x,x^2\}$ of $P_2$, you get precisely the vectors: $\{(a,b,c),(a_1,b_1,c_1),(a_2,b_2,c_2)\}$.

So $\{a + bx + cx^2, a_1 + b_1x + c_1x^2, a_2 + b_2x + c_2x^2\}$ form a basis if and only if $$\det\left( \begin{bmatrix} a & a_1 & a_2 \\ b & b_1 & b_2 \\ c & c_1 & c_2 \end{bmatrix}\right) \neq 0.$$

And this is true if and only if $\{(a,b,c),(a_1,b_1,c_1),(a_2,b_2,c_2)\}$ for a basis of $\mathbb{R}^3$.