Answer
$a.\quad $True
$b.\quad $True
$c.\quad $True
$d.\quad $False
$e.\quad $True
Work Step by Step
We compare the statements with
equivalent statements (a) to (l) of Theorem 8 and rewrite each.
When we write If (x) is true... we mean If statement (x) in Th.8 is true ...
$a.\quad $
If (k) is true then (j) is true ...
TRUE
$b.\quad $
If (e) is true then (h) is true ...
TRUE
$c.\quad $
If (g) is true then (f) is true ...
if $x \mapsto b$, b is unique because Ax is one-to-one.
TRUE
$d.\quad $
Ax mapping into $\mathbb{R}^{n}$ means that its image is not necessarily the whole $\mathbb{R}^{n}$,
so this statement says:
If (i) is not necessarily true, then (c) is true.
Since (c) and (i) are equivalent, this statement is FALSE
$e.\quad $
If there is a $\mathrm{b}$ in $\mathbb{R}^{n}$ such that the equation $A\mathrm{x} =\mathrm{b}$ is inconsistent, then (g) is false.
The statement reads:
If (g) is not true then (f) is not true ...
TRUE
