Answer
Not Necessarily.
Work Step by Step
The problem specifies that $Lx=0$ has the trivial solution, but it never specifies that $Lx=0$ has $only$ the trivial solution. Because all homogenous systems can have the trivial solution, the system $Lx=0$ could be either linearly dependent or linearly independent.
Therefore, we cannot make any statements about columns or span based on the information given; the columns of A do not necessarily span $R^{n}$.
$Note:$
Had the problem specified that $Lx=0$ has $only$ the trivial solution, statement (d) in Theorem 8 would be valid. With statement (d) valid, statement (h) in Theorem 8 could provide that the columns of L span $R^{n}$ .
