Chapter 3.SE, Problem 17SE

In Exercise $\text{12-18}$, $b={\left[a,b,c,d\right]}^{\text{T}}$, $T:R^6\to R^4$ is a linear transformation defined by $T\left(\text{x}\right)=A\text{x}$, and $A$ is a $\left(4\times 6\right)$ matrix such that the augmented matrix $\left[A|\text{b}\right]$ reduces to

$\left[\begin{array}{cccccc}1& 0& 2& 0& -3& 1\\ 0& 1& -1& 0& 2& 2\\ 0& 0& 0& 1& -1& -2\\ 0& 0& 0& 0& 0& 0\end{array}|\begin{array}{c}4a+b-2c\\ 12a+5b-7c\\ -5a-2b+3c\\ -16a-7b+9b+d\end{array}\right]$

a) Give an algebraic specification for $\Re \left(T\right)$, and use that specification to determine

a basis for $\Re \left(T\right)$.

b) Show that $\text{b}={\left[1,2,3,3\right]}^{\text{T}}$ is in $\Re \left(T\right)$, and express $\text{b}$ as a linear combination of the basis vectors found in part a).