Determine if its linear the transformation
$f:\Re_{3} [x]\rightarrow M_{2x2}$
such as
$f(ax^3+bx^2+cx+d) = \begin{bmatrix} a-c & 0 \\ 0&b+d \end{bmatrix}$
for any $ax^3+bx^2+cx+d \in \Re _{3} [x]$
Im having trouble associating the polynomial with the resulting matrix. I know I'm supposed to check the two main Linear transformation conditions but I don't know how to aproach this example.
Here's my attempt:
Condition 1: $T(u+v)=T(u)+T(v)$
$f((ax^3+bx^2+cx+d)+(a'x^3+b'x^2+c'x+d'))=f((ax^3+a'x^3)+(bx^2+b'x^2)+(cx+c'x)+(d+d'))=f((a+a')x^3+(b+b')x^2+(c+c')x+(d+d'))=\begin{bmatrix} (a+a')-(c+c') & 0\\ 0 & (b+b')+(d+d') \end{bmatrix} = \begin{bmatrix} a-c & 0\\ 0 & b+d \end{bmatrix} + \begin{bmatrix} a'-c' & 0\\ 0 & b'+d' \end{bmatrix}=f(ax^3+bx^2+cx+d)+f(a'x^3+b'x^2+c'x+d')$
Is this correct?
That looks fine. Another way to go is this:
Consider the standard basis $$\bigl\{A_{1,1},A_{2,1},A_{1,2},A_{2,2}\bigr\}$$ for $M_{2\times 2},$ where $A_{i,j}$ is the $2\times 2$ matrix of zeroes, except the $i$th row $j$th column entry, which is $1.$ Considering also the standard basis $\{x^3,x^2,x,1\}$ for $\mathfrak{R}_3[x],$ then the given transformation has the matrix representation $$\begin{bmatrix}1 & 0 & -1 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 1 & 0 & 1\end{bmatrix}.$$
Since it has a matrix representation, then it's a linear transformation.