Bergen

2020-10-20

In this problem, allow ${T}_{1}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$ and ${T}_{2}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$ be linear transformations. Find $Ker\left({T}_{1}\right),Ker\left({T}_{2}\right),Ker\left({T}_{3}\right)$ of the respective matrices:
$A=\left[\begin{array}{cc}1& -1\\ -2& 0\end{array}\right],B=\left[\begin{array}{cc}1& 5\\ -2& 0\end{array}\right]$

rogreenhoxa8

Step 1

$A=\left[\begin{array}{cc}1& -1\\ -2& 0\end{array}\right],B=\left[\begin{array}{cc}1& 5\\ -2& 0\end{array}\right]$ are matrices with respect to  Then if $x\in Ker\left({T}_{1}\right)$ Then ${T}_{1}\left(x\right)=0$ i.e. $Ax=0⇒\left[\begin{array}{cc}1& -1\\ -2& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$
${x}_{1}-{x}_{2}=0,-2{x}_{1}=0⇒{x}_{1}=0$
${x}_{1}={x}_{2}⇒{x}_{2}=0$
So $x=\left(\begin{array}{c}0\\ 0\end{array}\right)$ only so $Ker\left({T}_{1}\right)=\left\{\left(\begin{array}{c}0\\ 0\end{array}\right)\right\}$
Let $x\in Ker\left({T}_{2}\right)⇒{T}_{2}\left(x\right)=0$ and $Bx=0$
$⇒\left[\begin{array}{cc}1& 5\\ -2& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]⇒{x}_{1}+5{x}_{2}=0,-2{x}_{1}=0$

So $x=\left(\begin{array}{c}0\\ 0\end{array}\right)$
i.e. $Ker\left({T}_{2}\right)=\left\{\left(\begin{array}{c}0\\ 0\end{array}\right)\right\}$