Chapter 3.4, Problem 14E

In Exercises 11-16:

a) Find a matrix $B$ in reduced echelon form such that $B$ is row equivalent to the given matrix $A$.

b) Find a basis for the null space of $A$.

c) As in Example 6, find a basis for the range of $A$ that consists of columns of $A$. For each column, $A_j$ of $A$ that does not appear in the basis, express $A_j$ as a linear combination of the basis vectors.

d) Exhibit a basis for the row space of $A$.

$A=\left[\begin{array}{ccc}2& 2& 0\\ 2& 1& 1\\ 2& 3& 0\end{array}\right]$

EXAMPLE 6 Let $W$ be the subspace of $R^4$ spanned by the set $S=\left\{v_1,v_2,v_3,v_4,v_5\right\}$, where

$v_1=\left[\begin{array}{c}1\\ 1\\ 2\\ -1\end{array}\right]$, $v_2=\left[\begin{array}{c}1\\ 2\\ 1\\ 1\end{array}\right]$, $v_3=\left[\begin{array}{c}1\\ 4\\ -1\\ 5\end{array}\right]$, $v_4=\left[\begin{array}{c}1\\ 0\\ 4\\ -1\end{array}\right]$, and $v_5=\left[\begin{array}{c}2\\ 5\\ 0\\ 2\end{array}\right]$

Find a subset of $S$ that is a basis for $W$.