Chapter 4.6, Problem 41E

Rank, Nullity, Bases, and Linear Independence In Exercises $41$and $42$, use the fact that matrices $A$and $B$are row-equivalent.

(a) Find the rank and nullity of $A$.

(b) Find a basis for the nullspace of $A$.

(c) Find a basis for the row space of $A$.

(d) Find a basis for the column space of $A$.

(e) Determine whether the rows of $A$are linearly independent.

(f) Let the columns of $A$be denoted by $a_1,a_2,a_3,a_4$, and $a_5$.

Determine whether each set is linearly independent.

(i) $\left\{a_1,a_2,a_4\right\}$

(ii) $\left\{a_1,a_2,a_3\right\}$

(iii) $\left\{a_1,a_3,a_5\right\}$

$\begin{array}{l}A=\left[\begin{array}{ccccc}1& 2& 1& 0& 0\\ 2& 5& 1& 1& 0\\ 3& 7& 2& 2& -2\\ 4& 9& 3& -1& 4\end{array}\right]\\ B=\left[\begin{array}{ccccc}1& 0& 3& 0& -4\\ 0& 1& -1& 0& 2\\ 0& 0& 0& 1& -2\\ 0& 0& 0& 0& 0\end{array}\right]\end{array}$