Question

Quais das seguintes transformações são lineares?
a) T: R-------R²
    T (x,y,z) --------- (x-y,x,2z)


b) T: M2x2--------R
    T (A) = det A

Answer 1

Olá, Rogildo.

Uma transformação T é linear se satisfaz duas condições:

$\text{(i)}\,T(u+v)=T(u)+T(v)\\\\\text{(ii)}\,T( \alpha v)= \alpha T(v)$
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$\text{a)}\,\,T:\mathbb{R}^3\to\mathbb{R}^3,\text{ onde }T(x,y,z) =(x-y,x,2z),\text{ \' e linear, pois:}\\\\\text{(i)}\,T( \alpha x, \alpha y, \alpha z)=(\alpha x - \alpha y, \alpha x,2 \alpha z)\\\\ \alpha T(x,y,z) = \alpha (x-y,x,2z)=( \alpha x- \alpha y, \alpha x,2 \alpha z)\Rightarrow\\\\ T( \alpha x, \alpha y, \alpha z) = \alpha T(x,y,z)$


$\text{(ii)}\,T(x_1,y_1,z_1) =(x_1-y_1,x_1,2z_1)\\\\ T(x_2,y_2,z_2) =(x_2-y_2,x_2,2z_2)\\\\ T(x_1,y_1,z_1) +T(x_2,y_2,z_2)=(x_1-y_1,x_1,2z_1)+(x_2-y_2,x_2,2z_2)=\\\\ =(x_1-y_1+x_2-y_2,x_1+x_2,2z_1+2z_2)=\\\\=(x_1+x_2-(y_1+y_2),x_1+x_2,2(z_1+z_2))=\\\\ =T(x_1+x_2,y_1+y_2,z_1+z_2)$
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$\text{b)}\,\,T: M_{2\times2}\to\mathbb{R},\text{ onde }T(A)=\det A,\text{ n\~ao \' e linear, pois:}\\\\ \text{Seja }A= \left[\begin{array}{cc}a_{11}&a_{12}&\\a_{21}&a_{22}\end{array}\right] \Rightarrow \det A=a_{11}a_{22}-a_{12}a_{21}\\\\\\ \alpha A= \left[\begin{array}{cc} \alpha a_{11}& \alpha a_{12}&\\ \alpha a_{21}& \alpha a_{22}\end{array}\right] \Rightarrow$

$\underbrace{\det ( \alpha A)}_{T( \alpha A)}= \alpha ^2a_{11}a_{22}- \alpha ^2a_{12}a_{21}= \alpha ^2(a_{11}a_{22}-a_{12}a_{21})= \alpha ^2\underbrace{\det A\R}_{T(A)} \Rightarrow\\\\T( \alpha A)= \alpha^2 T(A)\Rightarrow T( \alpha A) \neq \alpha T(A)$
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Conclusão: a transformação da letra "a" é linear.