Chapter 3.SE, Problem 16SE

In Exercises 12-18, $b={\left[a,b,c,d\right]}^T$, $T:R^6\to R^4$ is a linear transformation defined by $T\left(x\right)=Ax$, and $A$ is a $\left(4\times 6\right)$ matrix such that augmented matrix $\left[\begin{array}{cc}A& b\end{array}\right]$ reduces to

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

Suppose that $A=\left[A_1,A_2,A_3,A_4,A_5,A_6\right]$.

a) For each vector $w_i,i=1,2,3,4,$ listed in Exercise $13$, if $w_i$ is in the column space of $A$, then express $w_i$ as a linear combination of the columns of $A$.

b) Find a subset of $\left\{A_1,A_2,A_3,A_4,A_5,A_6\right\}$ that is a basis for the column space of $A$.

c) For each column, $A_j$, of $A$ that does not appear in the basis obtained in part b), express $A_j$ as a linear combination of the basis vectors.

d) Let $b={\left[1,-2,1,-7\right]}^T$. Show that $b$ is in the column space of $A$, as a linear combination of the basis vectors found in part b).

e) If $x={\left[2,3,1,-1,1,1\right]}^T$, then express $Ax$ as a linear combination of the basis vectors found in part b).

$w_1=\left[\begin{array}{c}1\\ -1\\ 1\\ 0\end{array}\right]$, $w_2=\left[\begin{array}{c}1\\ 1\\ 3\\ 2\end{array}\right]$

$w_3=\left[\begin{array}{c}2\\ -2\\ 1\\ 9\end{array}\right]$ $w_4=\left[\begin{array}{c}2\\ 1\\ 4\\ 3\end{array}\right]$