Expert Answer to "Give a counterexample to show that the given..."

Suman Cole

Answered question

2021-11-08

Give a counterexample to show that the given transformation is not a linear transformation.
$T [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} x y \\ x + y \end{matrix}] T$

Answer & Explanation

ensojadasH

Skilled2021-11-09Added 100 answers

Step 1
Given,
The transformation
$T [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} x y \\ x + y \end{matrix}]$
Step 2
$T [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 3 \end{matrix}]$
$L e t v_{1} = [\begin{matrix} 1 \\ 2 \end{matrix}], v_{2} = [\begin{matrix} 0 \\ 1 \end{matrix}]$
Then,
$T v_{1} = T [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 2 \\ 3 \end{matrix}], T (v_{2}) = T [\begin{matrix} 0 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 1 \end{matrix}]$
$T v_{1} + T (v_{2}) = [\begin{matrix} 1 \\ 2 \end{matrix}] + [\begin{matrix} 0 \\ 1 \end{matrix}] = [\begin{matrix} 2 \\ 4 \end{matrix}] . . . (1)$
$T (v_{1} + v_{2}) = T ([\begin{matrix} 1 \\ 2 \end{matrix}] + [\begin{matrix} 0 \\ 1 \end{matrix}]) = T [\begin{matrix} 1 \\ 3 \end{matrix}] = [\begin{matrix} 3 \\ 4 \end{matrix}] . . . (2)$
On the other hand,
$T (v_{1} + v_{2}) \neq T (v_{1}) + T (v_{2})$

Hence the given transformation is not a linear transformation.

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