# In this problem, allow T_1: mathbb{R}^2 rightarrow mathbb{R}^2 and T_2: mathbb{R}^2 rightarrow mathbb{R}^2 be linear transformations. Find Ker(T_1), Ker(T_2), Ker(T_3) of the respective matrices:A=begin{bmatrix}1 & -1 -2 & 0 end{bmatrix} , B=begin{bmatrix}1 & 5 -2 & 0 end{bmatrix}

In this problem, allow $$T_1: \mathbb{R}^2 \rightarrow \mathbb{R}^2$$ and $$T_2: \mathbb{R}^2 \rightarrow \mathbb{R}^2$$ be linear transformations. Find $$Ker(T_1), Ker(T_2), Ker(T_3)$$ of the respective matrices:
$$A=\begin{bmatrix}1 & -1 \\-2 & 0 \end{bmatrix} , B=\begin{bmatrix}1 & 5 \\-2 & 0 \end{bmatrix}$$

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Step 1
$$T_1: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \text{ and } T_2: \mathbb{R}^2 \rightarrow \mathbb{R}^2$$
$$A=\begin{bmatrix}1 & -1 \\-2 & 0 \end{bmatrix} , B=\begin{bmatrix}1 & 5 \\-2 & 0 \end{bmatrix}$$ are matrices with respect to $$T_1 \text{ and } T_2$$ Then if $$x \in Ker(T_1)$$ Then $$T_1(x)=0$$ i.e. $$Ax=0 \Rightarrow \begin{bmatrix}1 & -1 \\-2 & 0 \end{bmatrix}\begin{bmatrix}x_1 \\x_2 \end{bmatrix}=\begin{bmatrix}0 \\0 \end{bmatrix}$$
$$x_1-x_2=0, -2x_1=0 \Rightarrow x_1=0$$
$$x_1=x_2 \Rightarrow x_2=0$$
So $$x=\begin{pmatrix}0 \\0 \end{pmatrix}$$ only so $$Ker(T_1)=\left\{\begin{pmatrix}0 \\0 \end{pmatrix}\right\}$$
Let $$x \in Ker(T_2) \Rightarrow T_2(x)=0$$ and $$Bx=0$$
$$\Rightarrow \begin{bmatrix}1 & 5 \\-2 & 0 \end{bmatrix}\begin{bmatrix}x_1 \\x_2 \end{bmatrix}=\begin{bmatrix}0 \\0 \end{bmatrix} \Rightarrow x_1+5x_2=0 , -2x_1=0$$
$$\Rightarrow x_1=0 \text{ and } x_2=0$$
So $$x=\begin{pmatrix}0 \\0 \end{pmatrix}$$
i.e. $$Ker(T_2)=\left\{\begin{pmatrix}0 \\0 \end{pmatrix}\right\}$$
Answer is as above