This was a question in one of my online assignments that I got wrong but have no idea why. Unfortunately I do not have the option to re-enter other attempts to see if they are correct, hence stack exchange.
Question:
Let p $=2x^2+6x+7$ .Find the coordinate vector of p relative to the following basis for $P_2$
$p_1 = 1$
$p_2 = 1+x$
$p_3 = 1+x+x^2$
My attempt:
Plugging in for simplicity $x=1$ for p , $p_1$, $p_2$, and $p_3$ results in the following vectors.
p = $(2,6,7)$
$p_1= $ $(1,0,0)$
$p_2= $ $(1,1,0)$
$p_3= $ $(1,1,1)$
Transposed of $p_n$ have:
$\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\end{bmatrix}$
*$\begin{bmatrix}7\\6\\2\end{bmatrix} = $$\begin{bmatrix}15\\8\\2\end{bmatrix}$
I reversed p because it was reversed relative to $p_n$
Best Answer
This is incorrect because you have simply taken the image of your polynomial under the matrix whose columns are the basis vectors.
Instead, you want to augment that same matrix with the vector you were given: $\left (\begin{array}{rrr|r}1&1&1&7\\0&1&1&6\\0&0&1&2\end{array}\right)$.
Now you can read off the solution. For starters, $z=2$.
You are looking for the linear combination of the basis vectors which will give you $(7,6,2)^t$.