Hello,
T: P2(R) -> P3(R) defined by T(f(x)) = xf(x) + f`(x).
Let A,B,C,a,b,c be elements of R and x,y elements of P2, where f(x) = (a + bx + cx^2) and f(y) = (A + By + Cy^2)
Can't I write this as a matrix vector product and prove it to be linear that way? I can also calculate the rank and null this way, right?
T(f(x)) = x(a + bx + cx^2) + b + 2x
= b + 2ax + bx^2 + cx^3
is this right so far? I really want to use the matrix vector product method though, but I'm having a hard time making it into a matrix, any advice?
Fernando Revilla is correct that the best way to prove something is a linear transformation is to show that it satisfies the definition of a linear transformation.
Since you specifically ask about writing it as a matrix, remember that a linear transformation from vector space U to vector space V depends on specific choices of (ordered) bases for U and V. Now, apply the linear transformation to each of the basis vector of U in turn, writing the result in terms of the basis for V. The coefficients will be the columns of the matrix.
Here, U is the function space P2 of quadratic polynomials and V is the function space P3 of cubic polynomials. The standard basis for the first is 1, x, and and for the second, 1, x, , and .
Applying the linear transformation to "1" we get . The first column of the matrix representation is .
Applying the linear transformation to "x" we get . The second column of the matrix representation is .
Applying the linear transformation to " " we get . The third column of the matrix representation is .
The matrix representing the linear transformation, using these bases in these orders, is
.