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Let u,v,w ∈ R3 be an LI set. Is u+v+w,u+wv,uw also an LI set?

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Answer to a math question Let u,v,w ∈ R3 be an LI set. Is u+v+w,u+wv,uw also an LI set?

Expert avatar
Jon
4.6
108 Answers
Para demostrar que el conjunto { u+v+w, u+w-v, u-w } es linealmente independiente, debemos encontrar la relación de dependencia lineal que haga que la combinación lineal de estos vectores sea igual al vector cero.

Supongamos que existen escalares a , b , y c no todos iguales a cero, tales que

a(u+v+w) + b(u+w-v) + c(u-w) = 0

Multiplicando y sumando los vectores, obtenemos:

(a+b+c)u + (a+b-c)w + (a-b)u = 0

Esto se traduce en el siguiente sistema de ecuaciones:

\begin{cases} a + b + c = 0 \ a + b - c = 0 \ a - b = 0 \end{cases}

Resolviendo este sistema, encontramos que la única solución es a = b = c = 0 . Por lo tanto, el conjunto { u+v+w, u+w-v, u-w } es linealmente independiente.

Entonces, sí es un conjunto linealmente independiente.

\textbf{Respuesta:} Sí.

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