 # Let A := 5 1 16 -3 [2 3 9 4 (a) Find a basis for Nul A. (b) Find a basis for Col A. chillywilly12a 2020-12-30 Answered
Let A := 5 1 16 -3 [2 3 9 4 (a) Find a basis for Nul A. (b) Find a basis for Col A.
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$A=\left[\begin{array}{cccc}1& -2& 1& -5\\ 5& 1& 16& -3\\ 2& 3& 9& 4\end{array}\right]$ PREF of $Ais\left[\begin{array}{cccc}1& 0& 3& -1\\ 0& 1& 1& 2\\ 0& 0& 0& 0\end{array}\right]$ $X\in \nu llspaceofAx=0$
$\sim \left[\begin{array}{cccc}1& 0& 3& -1\\ 0& 1& 1& 2\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]$
$x=a\left[\begin{array}{c}-3\\ -1\\ 1\\ 0\end{array}\right]+\left[\begin{array}{c}1\\ -2\\ 0\\ 1\end{array}\right]$ Basis for null space is

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

Column space A leading one is first non-zero entry in a row (In RREF) The column contains leading one are the pivots columns do the basis $\left\{\left[\begin{array}{c}1\\ 5\\ 2\end{array}\right],\left[\begin{array}{c}-2\\ 1\\ 3\end{array}\right]\right\}$