Transforming T(p(x))= x*p(x)

This shows how multiplication by $x$ can be performed in a polynomial vector space.

% Define transformation matrix
T = [0 0; 1 0; 0 1]

% Define a polynomial p(x) = a + bx. In B = {1,x}, it's [a;b]
syms a b
p = [a;b]

% Transform
T*p

% Which is 0 + ax + bx^2
T =

     0     0
     1     0
     0     1

 
p =
 
 a
 b
 
 
ans =
 
 0
 a
 b