Resolved: "Prove that the given transformation is a linear..."

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leviattan0pi

Answered question

2021-11-11

Prove that the given transformation is a linear transformation,
$T [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} x - y + z \\ 2 x + y - 3 z \end{matrix}]$

Answer & Explanation

Poul1963

Beginner2021-11-12Added 15 answers

Given
$T [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} x - y + z \\ 2 x + y - 3 z \end{matrix}]$ Definition
Let $T : V \to W$ be a lineal transformation then
$i) T (v_{1} + v_{2}) = T (v_{1}) + T (v_{2}), f or a l l v_{1}, v_{2} \in V$
$i i) T (a v) = a v, a b e a s c a l a r and v \in V$
Proof
$T [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} x - y + z \\ 2 x + y - 3 z \end{matrix}]$
$T (x, y, z) = (x - y + z, 2 x + y - 3 z)$
$L e t v_{1} = (x_{1}, y_{1}, z_{1}) and v_{2} = (x_{2}, y_{2}, z_{2})$
$i)$
$v_{1} + V_{2} = (x_{1} + x_{2}, y_{1} + y_{2}, z_{1} + z_{2})$
$T (v_{1} + v_{2}) = T (x_{1} + x_{2}, y_{1} + y_{2}, z_{1} + z_{2})$
$= ((x_{1} + x_{2}) - (y_{1} + y_{2}) + (z_{1} + z_{2}), 2 (x_{1} + x_{2}) + (y_{1} + y_{2}) - 3 (z_{1} + z_{2}))$
$= ((x_{1} - y_{1} + z_{1}) + (x_{2} - y_{2} + z_{2}), (2_{x_{1}} + y_{1} - 3_{z_{1}}) + (2_{x_{2}} + y_{2} - 3_{z_{2}})) = ((x_{1} - y_{1} + z_{1}$

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