So, here's the question that I'm trying to get to the answer to:
Let $f: P_3(F) \to P_4(F)$ be a linear map between two polynomial spaces defined as follows $f(t) \to (2-t)f(t)$. Determine the matrix associated with this linear map under the canonical bases of both vector spaces.
Let $f(t) = at^3+bt^2+ct+d$. After the transformation, we get:
$(2-t)f(t) = (2-t)(at^3+bt^2+ct+d) = -at^4+(2a-b)t^3+(2b-c)t^2+(2c-d)t+2d$
If we treat a polynomial such as $f(t) = at^3+bt^2+ct+d$ as being entirely described by the vector $[a,b,c,d]$, then $f$ is the following transformation:
$f[a,b,c,d] = [-a,2a-b,2b-c,2c-d,2d]$
To obtain the columns of the matrix, we just need to find the images of the canonical basis of $P_3(F)$. Doing that, we get:
$f[1,0,0,0] = [-1,2,0,0,0] = Col_1$
$f[0,1,0,0] = [0,1,-2,0,0] = Col_2$
$f[0,0,1,0] = [0,0,1,-2,0] = Col_3$
$f[0,0,0,1] = [0,0,0,1,-2] = Col_4$
That gives us the matrix associatedd with this linear transformation.
So, my question is if the argument above is correct or not? If it isn't, how could I correct it?