Escriba el vector w = (1, 1, 1) como una combinación lineal de vectores en el conjunto : S{(1, 2, 3), (0, 1, 2), ( - 1, 0, 1)}?

Buecas hacer una combinación lineal entre los tres vectores para obtener el vector w, entonces, supongamos escalares <img src="https://tex.z-dn.net/?f=%5Calpha%2C%5Cbeta%2C%5Cgamma%5Cin%5CRe" />, entonces

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%20w%3D%5Calpha%281%2C2%2C3%29%2B%5Cbeta%280%2C1%2C2%29%2B%5Cgamma%28-1%2C0%2C1%29%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%5C%5C1%5C%5C1%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Calpha%5C%5C2%5Calpha%5C%5C3%5Calpha%5Cend%7Barray%7D%5Cright%5D%20%2B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%5C%5C%5Cbeta%5C%5C2%5Cbeta%5Cend%7Barray%7D%5Cright%5D%2B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-%5Cgamma%5C%5C0%5C%5C2%5Cgamma%5Cend%7Barray%7D%5Cright%5D%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%26-1%5C%5C2%261%260%5C%5C3%262%262%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Calpha%5C%5C%5Cbeta%5C%5C%5Cgamma%5Cend%7Barray%7D%5Cright%5D" />

entonces, el problema se reduce a resolver un sistema de ecuaciones, que podemos hacerlo usando Gauus - Jordan,

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%26-1%5C%5C2%261%260%5C%5C3%262%262%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Calpha%5C%5C%5Cbeta%5C%5C%5Cgamma%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%5C%5C1%5C%5C1%5Cend%7Barray%7D%5Cright%5D" />

entonces,

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cleft%5B%5Cbegin%7Barray%7D%7Brrr%7Cr%7D1%260%26-1%261%5C%5C2%261%260%261%5C%5C3%262%262%261%5Cend%7Barray%7D%5Cright%5D%5Cunderbrace%7B%5Clongrightarrow%7D_%7B%7B%5Csubstack%7BF_%7B2%7D-2F_%7B1%7D%5C%5CF_%7B3%7D-3F_%7B1%7D%7D%7D%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Brrr%7Cr%7D1%260%26-1%261%5C%5C0%261%262%26-1%5C%5C0%262%265%26-2%5Cend%7Barray%7D%5Cright%5D%5Cunderbrace%7B%5Clongrightarrow%7D_%7BF_%7B3%7D-2F_%7B2%7D%7D%3D%5C%5C%5C%5C%5Cleft%5B%5Cbegin%7Barray%7D%7Brrr%7Cr%7D1%260%26-1%261%5C%5C0%261%262%26-1%5C%5C0%260%261%260%5Cend%7Barray%7D%5Cright%5D%5Cunderbrace%7B%5Clongrightarrow%7D_%7B%7B%5Csubstack%7BF_%7B2%7D-2F_%7B3%7D%5C%5CF_%7B1%7D%2BF_%7B3%7D%7D%7D%7D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Brrr%7Cr%7D1%260%260%261%5C%5C0%261%260%26-1%5C%5C0%260%261%260%5Cend%7Barray%7D%5Cright%5D" />

finalmente ya tenemos los vlores de los escalares, entonces comprobemos,

<img src="https://tex.z-dn.net/?f=%281%2C1%2C1%29%3D%281%29%281%2C2%2C3%29%2B%28-1%29%280%2C1%2C2%29%2B%280%29%28-1%2C0%2C1%29%3D%5C%5C...%3D%281%2C2%2C3%29%2B%280%2C-1%2C-2%29%3D%281%2C1%2C1%29" />

y como ves, efectivamente esas constante fomar combinación lineal del vector w, además podemos decir que son linealmente dependientes.

Y eso sería todo.