Find the transformation matrix: $$F: \mathbb R_3[x]\ \mathbb ] \to \mathbb R_3[x] $$ $$F(v) = \frac{d^2 v}{dv^2}$$ Basis: $1, x, x^2, x^3$ and $\mathbb R_3[x]$ - 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 $$\frac{d^2y}{dy^2}=2+6x$$ 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 $$\begin{bmatrix} 0&0&2&0 \\ 0 &0&0&6 \\ 0&0&0&0 \\ 0&0&0&0 \end{bmatrix} $$ Or maybe $$ \begin{bmatrix} 2&0&0&0 \\ 0&6&0&0 \\ 0&0 &0 &0 \\0 &0 &0 &0 \end{bmatrix}$$ Because both of them, when multiplied by \begin{bmatrix} 1 \\ x \\ x^2 \\ x^3 \end{bmatrix} Will give the correct answer. Thus, what is the correct way to solve this?
If you pick an ordered basis for the domain and co domain then the order is fixed.
Pick the basis $x\mapsto 1, x\mapsto x, x\mapsto x^2, x\mapsto x^3$.