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}

Bergen 2020-10-20 Answered

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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Expert Answer

rogreenhoxa8
Answered 2020-10-21 Author has 25473 answers

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

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