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)}
seeker17:
Gracias, dentro de un rato ya te ayudo
Respuestas
Respuesta dada por:
5
Buecas hacer una combinación lineal entre los tres vectores para obtener el vector w, entonces, supongamos escalares
, entonces
![\displaystyle w=\alpha(1,2,3)+\beta(0,1,2)+\gamma(-1,0,1)\\\left[\begin{array}{ccc}1\\1\\1\end{array}\right] =\left[\begin{array}{ccc}\alpha\\2\alpha\\3\alpha\end{array}\right] +\left[\begin{array}{ccc}0\\\beta\\2\beta\end{array}\right]+\left[\begin{array}{ccc}-\gamma\\0\\2\gamma\end{array}\right]= \left[\begin{array}{ccc}1&0&-1\\2&1&0\\3&2&2\end{array}\right]\left[\begin{array}{ccc}\alpha\\\beta\\\gamma\end{array}\right] \displaystyle w=\alpha(1,2,3)+\beta(0,1,2)+\gamma(-1,0,1)\\\left[\begin{array}{ccc}1\\1\\1\end{array}\right] =\left[\begin{array}{ccc}\alpha\\2\alpha\\3\alpha\end{array}\right] +\left[\begin{array}{ccc}0\\\beta\\2\beta\end{array}\right]+\left[\begin{array}{ccc}-\gamma\\0\\2\gamma\end{array}\right]= \left[\begin{array}{ccc}1&0&-1\\2&1&0\\3&2&2\end{array}\right]\left[\begin{array}{ccc}\alpha\\\beta\\\gamma\end{array}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle+w%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+%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%5Calpha%5C%5C2%5Calpha%5C%5C3%5Calpha%5Cend%7Barray%7D%5Cright%5D+%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++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B0%26amp%3B-1%5C%5C2%26amp%3B1%26amp%3B0%5C%5C3%26amp%3B2%26amp%3B2%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,
![\displaystyle \left[\begin{array}{ccc}1&0&-1\\2&1&0\\3&2&2\end{array}\right]\left[\begin{array}{ccc}\alpha\\\beta\\\gamma\end{array}\right]=\left[\begin{array}{ccc}1\\1\\1\end{array}\right] \displaystyle \left[\begin{array}{ccc}1&0&-1\\2&1&0\\3&2&2\end{array}\right]\left[\begin{array}{ccc}\alpha\\\beta\\\gamma\end{array}\right]=\left[\begin{array}{ccc}1\\1\\1\end{array}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B0%26amp%3B-1%5C%5C2%26amp%3B1%26amp%3B0%5C%5C3%26amp%3B2%26amp%3B2%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,
![\displaystyle\left[\begin{array}{rrr|r}1&0&-1&1\\2&1&0&1\\3&2&2&1\end{array}\right]\underbrace{\longrightarrow}_{{\substack{F_{2}-2F_{1}\\F_{3}-3F_{1}}}}=\left[\begin{array}{rrr|r}1&0&-1&1\\0&1&2&-1\\0&2&5&-2\end{array}\right]\underbrace{\longrightarrow}_{F_{3}-2F_{2}}=\\\\\left[\begin{array}{rrr|r}1&0&-1&1\\0&1&2&-1\\0&0&1&0\end{array}\right]\underbrace{\longrightarrow}_{{\substack{F_{2}-2F_{3}\\F_{1}+F_{3}}}}=\left[\begin{array}{rrr|r}1&0&0&1\\0&1&0&-1\\0&0&1&0\end{array}\right] \displaystyle\left[\begin{array}{rrr|r}1&0&-1&1\\2&1&0&1\\3&2&2&1\end{array}\right]\underbrace{\longrightarrow}_{{\substack{F_{2}-2F_{1}\\F_{3}-3F_{1}}}}=\left[\begin{array}{rrr|r}1&0&-1&1\\0&1&2&-1\\0&2&5&-2\end{array}\right]\underbrace{\longrightarrow}_{F_{3}-2F_{2}}=\\\\\left[\begin{array}{rrr|r}1&0&-1&1\\0&1&2&-1\\0&0&1&0\end{array}\right]\underbrace{\longrightarrow}_{{\substack{F_{2}-2F_{3}\\F_{1}+F_{3}}}}=\left[\begin{array}{rrr|r}1&0&0&1\\0&1&0&-1\\0&0&1&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cleft%5B%5Cbegin%7Barray%7D%7Brrr%7Cr%7D1%26amp%3B0%26amp%3B-1%26amp%3B1%5C%5C2%26amp%3B1%26amp%3B0%26amp%3B1%5C%5C3%26amp%3B2%26amp%3B2%26amp%3B1%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%26amp%3B0%26amp%3B-1%26amp%3B1%5C%5C0%26amp%3B1%26amp%3B2%26amp%3B-1%5C%5C0%26amp%3B2%26amp%3B5%26amp%3B-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%26amp%3B0%26amp%3B-1%26amp%3B1%5C%5C0%26amp%3B1%26amp%3B2%26amp%3B-1%5C%5C0%26amp%3B0%26amp%3B1%26amp%3B0%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%26amp%3B0%26amp%3B0%26amp%3B1%5C%5C0%26amp%3B1%26amp%3B0%26amp%3B-1%5C%5C0%26amp%3B0%26amp%3B1%26amp%3B0%5Cend%7Barray%7D%5Cright%5D)
finalmente ya tenemos los vlores de los escalares, entonces comprobemos,
![(1,1,1)=(1)(1,2,3)+(-1)(0,1,2)+(0)(-1,0,1)=\\...=(1,2,3)+(0,-1,-2)=(1,1,1) (1,1,1)=(1)(1,2,3)+(-1)(0,1,2)+(0)(-1,0,1)=\\...=(1,2,3)+(0,-1,-2)=(1,1,1)](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
entonces, el problema se reduce a resolver un sistema de ecuaciones, que podemos hacerlo usando Gauus-Jordan,
entonces,
finalmente ya tenemos los vlores de los escalares, entonces comprobemos,
y como ves, efectivamente esas constante fomar combinación lineal del vector w, además podemos decir que son linealmente dependientes...y eso sería todo
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