 Mylo O'Moore

2021-03-06

Determine the null space of each of the following matrices:
$\left(\begin{array}{ccc}1& 3& -4\\ 2& -1& -1\\ -1& -3& 4\end{array}\right)$ SoosteethicU

Step 1
Look at the matrices provided:
$\left(\begin{array}{ccc}1& 3& -4\\ 2& -1& -1\\ -1& -3& 4\end{array}\right)$
To discover the null spaces, first solve the following system of equations:
$\left(\begin{array}{ccc}1& 3& -4\\ 2& -1& -1\\ -1& -3& 4\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$
Use the Gauss Elimination method to solve the aforementioned matrix.  Find the row echelon form: ${R}_{2}={R}_{2}-2{R}_{1}$
$\left(\begin{array}{ccc}1& 3& -4\\ 0& -7& 7\\ -1& -3& 4\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$
${R}_{3}={R}_{3}+{R}_{1}$
$\left(\begin{array}{ccc}1& 3& -4\\ 0& -7& 7\\ 0& 0& 0\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$
Step 2
we get equations:
$-7{x}_{2}+7{x}_{3}=0$
$7{x}_{2}=7{x}_{3}$
${x}_{2}={x}_{3}$ And ${x}_{1}+3{x}_{2}-4{x}_{3}=0$
${x}_{1}+3{x}_{2}-4{x}_{2}=0$
${x}_{1}={x}_{2}$
Therefore null space is given by:
$\left(\begin{array}{c}{x}_{2}\\ {x}_{2}\\ {x}_{2}\end{array}\right)={x}_{2}\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$ Jeffrey Jordon

Answer is given below (on video) Jeffrey Jordon