Finding basis for $P_2$ that satisfy conditions

Question

I want to find a basis $\mathbb{B}$ such that

$[P_2]_\mathbb{B} = \begin{bmatrix} p(0) \\ p(1) \\ p(2) \end {bmatrix}$

I assume general $P_2 = a+bx+cx^2$

Since $p(0)=a$

$p(1) = a+b+c$,

$p(2)=a+2b+4c$

I want to find a basis $\mathbb{B}$ such that $[P_2]_\mathbb{B} = \begin{bmatrix} a \\ a+b+c \\ a+2b+4c \end {bmatrix}$

I tried $[{1}, {1+x+x^2}, 1+2x+4x^2]$. I just tried using the general basis for $P_2$: $[1, x, x^2]$ and logically deducing the basis that will satisfy the conditions in the task. Is the basis I've arrived at correct?

Answer 1

Goal: Find a basis $\mathbb{B}$ such that $$ [p]_\mathbb{B} = \begin{bmatrix} p(0) \\ p(1) \\ p(2) \end{bmatrix}$$ for every $p \in P_2$.

Let $p = a + bx + cx^2$ be an arbitrary element in $P_2$ with coefficients $a,b,c$. If we consider the standard basis $\mathbb{E} = \{ 1, x, x^2 \}$, we get $$ [p]_\mathbb{E} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}.$$

On the one hand, as you showed us, we are looking for a basis $\mathbb{B}$ such that $$ [p]_\mathbb{B} = \begin{bmatrix} a \\ a+b+c \\ a+2b+4c \end{bmatrix}$$.

But on the other hand, we know that $$ T_{\mathbb{E},\mathbb{B}} [p]_\mathbb{E} = [p]_\mathbb{B} $$ where $T_{\mathbb{E},\mathbb{B}}$ the basis change matrix from $\mathbb{E}$ to $\mathbb{B}$.

As it turns out, we must have $$T_{\mathbb{E},\mathbb{B}} = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{bmatrix}$$

whose inverse is $$ T_{\mathbb{B},\mathbb{E}} = \begin{bmatrix} 1 & 0 & 0 \\ -\frac{3}{2} & 2 & -\frac{1}{2} \\ \frac{1}{2} & -1 & \frac{1}{2} \end{bmatrix} $$ which is equivalent to $\mathbb{B}$ being the basis $\left[1 -\frac{3}{2}x + \frac{1}{2}x^2 , 2x-x^2, -\frac{1}{2}x+\frac{1}{2}x^2 \right]$.

Answer 2

Starting out from the standard basis is good.
Calling $P_2$ the general element of $P_2$ and refer to the same element as $p$ in the next line is not good. We can calmly evaluate polynomials, i.e. if $p\in P_2$, then e.g. $p(1)$ makes perfect sense. So you just wanted to write $[p]_{\Bbb B}$.

Hint: We can get the coordinates of the standard basis vectors in the new basis. Write them up in the columns of a matrix and invert that matrix.