 # Find the transformation matrix: F : <mi mathvariant="double-struck">R 3 < kokoszzm 2022-06-30 Answered
Find the transformation matrix:

$F\left(v\right)=\frac{{d}^{2}v}{d{v}^{2}}$
Basis: $1,x,{x}^{2},{x}^{3}$ and ${\mathbb{R}}_{3}\left[x\right]$ - the set of all third degree polynomials of variable $x$ over $\mathbb{R}$ Assume that all coefficients of the polynomials are $1$
The first thing that springs to my mind is to calculate this derivative by hand, and so we got
$\frac{{d}^{2}y}{d{y}^{2}}=2+6x$
Now, we need to put these values - $2$ and $6$ in such a matrix that - when multiplied by the basis vector -will give us $2+6x$ But there are many ways I can think of, for example
$\left[\begin{array}{cccc}0& 0& 2& 0\\ 0& 0& 0& 6\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
Or maybe
$\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 6& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
Because both of them, when multiplied by$\left[\begin{array}{c}1\\ x\\ {x}^{2}\\ {x}^{3}\end{array}\right]$
Will give the correct answer. Thus, what is the correct way to solve this?
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If you pick an ordered basis for the domain and co domain then the order is fixed.
Pick the basis $x↦1,x↦x,x↦{x}^{2},x↦{x}^{3}$

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