Question

Álgebra Linear

Verificar se u e v são linearmente dependente ou linearmente independente nos casos:

a) u=(1,2) e v=(3,6)
b) u=(4,-6) e v=(-2,3)
c) u=(6,8) e v=(-2,-3)
d) u=(0,0) e v=(1,5)

Answer 1

Olá


Iguala as coordenadas dos vetores

A)

$\displaystyle \vec{u}=(1,2)\\\vec{v}=(3,6)\\\\1=3\alpha~~~~~ \Longrightarrow ~~~~ \boxed{\alpha= \frac{1}{3}} \\\\2=6\alpha~~~~~ \Longrightarrow ~~~~\alpha= \frac{2}{6}~~~~~\Rightarrow \boxed{\alpha = \frac{1}{3} }$

Vetores L.D


B)

$\displaystyle \vec{u}=(4,-6)\\\vec{v}=(-2,3)\\\\4=-2\alpha~~~~~ \Longrightarrow ~~~~ \alpha= \frac{4}{-2} ~~~~\Rightarrow \boxed{\alpha=-2}\\\\-6=3\alpha~~~~~ \Longrightarrow ~~~~\alpha= ~~~~\frac{-6}{3}~~~~~\Rightarrow \boxed{\alpha = -2 }$

Vetores L.D


C)

$\displaystyle \vec{u}=(6,8)\\\vec{v}=(-2,-3)\\\\6=-2\alpha~~~~~ \Longrightarrow ~~~~ \boxed{\alpha=-3} \\\\8=-3\alpha~~~~~\Rightarrow \boxed{\alpha = -\frac{8}{3} }$

Vetores L.I


D)

$\displaystyle \vec{u}=(0,0)\\\vec{v}=(1,5)\\\\0=1\alpha~~~~~ \Longrightarrow ~~~~ \boxed{\alpha= 0} \\\\0=5\alpha~~~~~ \Rightarrow \boxed{\alpha = 0} }$