Find 2 by 3 matrix M such that M $\begin{pmatrix}a \\ b \\ c\end{pmatrix}= \begin{pmatrix}d \\ e\end{pmatrix}$ whenever $(ax^2+bx + c)'=dx+e$

Question

Find 2 by 3 matrix M such that M $\begin{pmatrix}a \\ b \\ c\end{pmatrix}= \begin{pmatrix}d \\ e\end{pmatrix}$ whenever $(ax^2+bx + c)'=dx+e$.

Here is my take on it. \begin{align*}(ax^2+bx+c)' = 2ax +b = dx+e\end{align*} which gives $a = \frac{d}{2}$ and $b = e$. So we have, \begin{align*}\begin{pmatrix}m_1 & m_2 & m_3 \\ m_4 & m_5 & m_6\end{pmatrix}\begin{pmatrix}a \\ b \\ c\end{pmatrix} = \begin{pmatrix}am_1+bm_2+cm_3\\ am_4 + bm_5 + cm_6 \end{pmatrix}=\begin{pmatrix}2a \\ b\end{pmatrix}\end{align*}After solving the above equation, I got $m_1 +m_4 = 2, m_2 + m_5 = 1$ and $m_3 + m_6 = 0$.

I can just set $m_1, m_2$ and $m_3$ to be free variable but I don't think this is the right answer. And I am not sure if I am going to the right direction either. It will be really helpful for me if someone can point me to the right direction.

Answer 1

You have $6$ variables and $2$ constraints. Therefore you will have $4$ free variables... giving you a $4$-dimensional solution space. Choose $m_6, m_5,m_3$ and $m_2$ freely, and back substitute to get $m_1$ and $m_2$...

Or, perhaps better,why not use $m_1=2$ and $m_5=1$, then set the other $m$'s to $0$?

Answer 2

The columns of $M$ are images of the standard basis vectors, which correspond to the polynomials $x^2$, $x$ and $1$, in that order. $(x^2)'=2x$, so the first column of $M$ is $(2,0)^T$. Continue in this way with $x$ and $1$ to find the second and third columns, respectively, of $M$.