[tex]v=(1,-3)\\ w=(-4,5)\\ u=(1,-2)[/tex]
[tex]\boxed{u=\lambda v+\beta w}[/tex]
É pedido para determinar dois números [tex]\lambda\in\mathbb{R}[/tex] e [tex]\beta\in\mathbb{R}[/tex] tal que u seja combinação linear de v e w. Logo:
[tex]u=\lambda v+\beta w\ \therefore\ (1,-2)=\lambda(1,-3)+\beta(-4,5)\ \therefore\\\\ (1,-2)=(\lambda-4\beta,-3\lambda+5\beta)[/tex]
Dessa forma, é possível formar um sistema linear com duas expressões e duas incógnitas:
[tex]$\left\{\begin{array}{cc}\lambda-4\beta=1\\-3\lambda+5\beta=-2\end{array}\right$[/tex]
[tex]\lambda=4\beta+1\ \therefore\ -3(4\beta+1)+5\beta=-2\ \therefore\\\\ -12\beta-3+5\beta=-2\ \therefore\ -7\beta=1\ \therefore\ \boxed{\beta=-\dfrac{1}{7}}[/tex]
[tex]\lambda=4\bigg(-\dfrac{1}{7}\bigg)+1=-\dfrac{4}{7}+\dfrac{7}{7}\ \therefore\ \boxed{\lambda=\dfrac{3}{7}}[/tex]
Dessa forma, a combinação linear de v e w que resulta em u é:
[tex](1,-2)=\dfrac{3}{7}(1,-3)-\dfrac{1}{7}(-4,5)\ \therefore\ \boxed{u=\dfrac{3}{7}v-\dfrac{1}{7}w}[/tex]
