Find a vector $t \in \{x,y,z\}$ with base $\{u, v, w\}$ I don't know how to find a vector $\vec t$ that will suffice the condition:
$\vec t \in \{x,y,z\}$ with bases $\{u, v, w\}$
the given vectors are:
$$
\begin{array}{rcrrrrrl} u &=& [ & -3, & -1, & 1, & -2 & ] \\ v &=& [ & 2, & -3, & -2, & -2 & ] \\ w &=& [ & 1, & 1, & 1, & -1 & ] \\ x &=& [ & 7, & -4, & 1, & -5 & ] \\ y &=& [ & -10, & -2, & 4, & 2 & ] \\ z &=& [ & -7, & 2, & 5, & -3 & ] \\ \end{array}
$$
I need to find the linear combination $t = \alpha u + \beta v + \gamma w $ that will fulfill the condition.
Thank you a lot!

I am not sure, whether I have understood your problem correctly. If you have to say whether $x$, $y$ or $z$ are in $\operatorname{span}\{u,v,w\}$ you have to solve the three linear systems:


*

*$x = \alpha u + \beta v + \gamma w$

*$y = \alpha u + \beta v + \gamma w$

*$z = \alpha u + \beta v + \gamma w$


If there is a solution for any of the vectors $x$, $y$ or $z$, then these vectors are in $\operatorname{span}\{u,v,w\}$ and the computed vector $(\alpha, \beta, \gamma)$ is the representation of the vector in the base $\{u,v,w\}$. If there is no solution for a linear system, then the regarding vector $x$, $y$ or $z$ is not in $\operatorname{span}\{u,v,w\}$.
