Chapter 6.CR, Problem 102CR

To determine

(a)

The given statement “For polynomials, the differential operator $D_x$ is a linear transformation from $P_n$ into $P_{n-1}$.” is true or false, if true provide the appropriate statement and if false provide the appropriate example.

To determine

(b)

The given statement “The set of all vectors v in V that satisfy $T\left(\text{v}\right)=\text{v}$ is the kernel of T.” is true or false, if true provide the appropriate statement and if false provide the appropriate example.

To determine

(c)

The given statement “The standard matrix A of the composition of two linear transformation $T\left(\text{v}\right)=T_2\left(T_1\left(\text{v}\right)\right)$ is the product of the standard matrix for $T_2$ and the standard matrix for $T_1$.” is true or false, if true provide the appropriate statement and if false provide the appropriate example.