Chapter 5.SE, Problem 11SE

In Exercise 7-11, Use the fact that the matrix

$\left[A|b\right]=\left[\begin{array}{cccccc}1& -1& 3& 1& 3& 2\\ 1& 0& 2& 3& 2& 3\\ 0& -2& 2& -4& 3& 0\\ 2& -1& 5& 4& 6& 7\end{array}|\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]$

is row equivalent to

$\left[\begin{array}{cccccc}1& 0& 2& 3& 0& -1\\ 0& 1& -1& 2& 0& 3\\ 0& 0& 0& 0& 1& 2\\ 0& 0& 0& 0& 0& 0\end{array}|\begin{array}{c}4a-3b-2c\\ -3a+3b+c\\ -2a+2b+c\\ a-3b-c+d\end{array}\right]$.

Let $V$ be the vector space for all $\left(2\times 3\right)$ matrices, and suppose that $T:V\to {\mathcal{P}}_3$ is the linear transformation defined by,

$\begin{array}{l}T\left(\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\end{array}\right]\right)\\ =\left(a_{11}-a_{12}+3a_{13}+a_{21}+3a_{22}+2a_{23}\right)\\ +\left(a_{11}+2a_{13}+3a_{21}+2a_{22}+3a_{23}\right)x\\ +\left(-2a_{12}+2a_{13}-4a_{21}+3a_{22}\right)x^2\\ +\left(2a_{11}-a_{21}+5a_{13}+4a_{21}+6a_{22}+7a_{23}\right)x^3\end{array}$

a. Calculate the matrix of $T$ relative to the natural bases $B$ and $C$ for $V$ and ${\mathcal{P}}_3$, respectively.

b. Determine the rank and the nullity of $T$.

c. Give an algebraic specification to determine a basis for $\mathcal{R}\left(T\right)$ and use the specification to determine a basis for $\mathcal{R}\left(T\right)$.

d. Show that $q\left(x\right)=1+2x-x^2+4x^3$ is in $\mathcal{R}\left(T\right)$ and find a matrix $A$ in $V$ such that $T\left(A\right)=q\left(x\right)$.

e. Find a basis for $\mathcal{N}\left(T\right)$.