[SOLVED] Math 231 — hw 18

1. In class, we discussed the differential operator, D, over the space P3, the space of polynomials up to degree 3. The differential operator takes polynomials to their derivatives.
Solve the following equation: D(ax3 + bx2 + cx + d) =
2. If ax3 + bx2 + cx + d is represented as the column vector


a
b
c
d

 ,
write out M(D). (Hint: Use your previous answer.)
3. In class, we stated that the null D is the space of constant functions. What is the
representation of this null space? In other words, what is null M(D)?
4. Suppose S is a map that represents a shift in vectors over R
3
. S(a, b, c) = (b, c, 0).
Describe its null space and give a representation M(S).
5. Now suppose we define a function P that represents a permutation over the vector
space R
3
. P(a, b, c) = (b, c, a). Describe its null space and give a representation M(P).