Exercise 10

Example 2.11 gives a matrix that is singular and is therefore associated with maps that are singular. We cannot state the action of the associated map $g$ on domain elements $\vec{v}\in V$, because do not know the domain $V$ or codomain $W$ or the starting and ending bases $B$ and $D$. But we can compute what happens to the representations ${\rm Rep}_{B,D}(\vec{v})$.

• Find the set of column vectors representing the members of the null space of any map $g$ represented by this matrix.

• Find the nullity of any such map $g$.

• Find the set of column vectors representing the members of the range space of any map $g$ represented by the matrix.

• Find the rank of any such map $g$.

• Check that rank plus nullity equals the dimension of the domain.