Question

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15) Consideremos a transformação linear 2 2 T :ℜ → ℜ definida por
T x y x y x y ( , ) (3 2 , 4 ) = − + . Utilizar os vetores u = (1,2) e v = − (3, 1) para mostrar que
T u v T u T v (3 4 ) 3 ( ) 4 ( ) + = + .

Answer 1

Bom dia Diego!

Solução!

Seja os vetores!

$\vec{u}=(1,2)\\\\\ \vec{v}=(3,-1)$

Fazendo!

$\vec{3u}=(1,2)\\\\\ \vec{4v}=4(3,-1)$

$\vec{3u} =(3,6)\\\\\ \vec{4v} =(12,-4)$

$\vec{3u}+\vec{4v}=[(3,6)+(12,-4)]\\\\\\ \vec{3u}+\vec{4v}=[3+12,6-4]\\\\\\ \vec{3u}+\vec{4v}=[15,-2]$

Resultado 1

$T(\vec{3u}+\vec{4v})=(15,-2)= (3\times15-2\times2,15+2\times4)\\\\\\ T(\vec{3u}+\vec{4v})=(15,-2) = (45-4,15+8)\\\\\\ T(\vec{3u}+\vec{4v})=(15,-2) =(41,23)$


$T(u)=T(1,2)=(-1,9)~~ Logo~~3T(u)=(-3,27)\\\\\\\ T(v)=T(3,-1)=(11,-1)~~logo~~4T(v)=(44,-4)$

Então!

$3T(u)+4T(v)=(41,23)=T(3(u)+4(v))\\\\\\\\\ Esta~~provado$

Bom dia!
Bons estudos!