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Question Number 215525 by zetamaths last updated on 09/Jan/25

$$\mathrm{\Phi }:{\mathbb{R}}^2\to {\mathbb{R}}^2$$$$(x.;y)\mid \to (2x+3y:3y)$$$$find$$$$$$$$$$$$$$$$\mathrm{\Phi }\in \mathcal{L}\left({\mathbb{R}}^2\right)$$$$findker\left(\mathrm{\Phi }\right)andtheim\left(\mathrm{\Phi }\right)$$$$fof=?$$$$$$

Answered by MrGaster last updated on 09/Jan/25

$$\ker \left(\mathrm{\Phi }\right)=\{x,y\}\in {\mathbb{R}}^2\mid \mathrm{\Phi }(x,y)=(0,0)\}$$$$\left(\begin{array}{cc}2& 3\\ 0& 3\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$$$$\{\begin{array}{l}2x+3y=0\\ 3y=0\end{array}$$$$y=0\Rightarrow 2x=0\Rightarrow x=0$$$$\mathrm{kel}\left(\mathrm{\Phi }\right)=\left\{(0,0)\right\}✓$$$$\mathrm{im}\left(\mathrm{\Phi }\right)=\{\mathrm{\Phi }(x,y)\mid (x,y)\in {\mathbb{R}}^2\}$$$$\mathrm{im}\left(\mathrm{\Phi }\right)=\mathrm{sqan}(\left(\begin{array}{c}2\\ 0\end{array}\right),\left(\begin{array}{c}3\\ 3\end{array}\right))$$$$\mathrm{Reduse}\mathrm{to}\mathrm{echelon}\mathrm{form}:$$$$\left(\begin{array}{cc}2& 3\\ 0& 3\end{array}\right)\sim \left(\begin{array}{cc}2& 0\\ 0& 3\end{array}\right)$$$$\mathrm{im}\left(\mathrm{\Phi }\right)=\mathrm{sqan}((\left(\begin{array}{c}2\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 3\end{array}\right))$$$$=\mathrm{sqan}(\left(\begin{array}{c}1\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 1\end{array}\right))$$$$={\mathbb{R}}^2✓$$$$\mathrm{Conclusion}:$$$$\ker \left(\mathrm{\Phi }\right)=\left\{(0,0)\right\}$$$$\mathrm{im}\left(\mathrm{\Phi }\right)={\mathbb{R}}^2$$$$f=2$$

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