if $\displaystyle \{u,v,w,\}$ is a linearly independent set of vectors, then $\displaystyle S = \{u+v-2w,u-v-w,u+w\}$ is also linearly independent.
True or false?
thanks
At first I set up the matrix with each vector in a row like this:
$\displaystyle
\left(
\begin{matrix}
u & v & -2w\\
u & -v & -w\\
u & 0 & w\\
\end{matrix}
\right)
$
But then I couldn't see how this would work. Then I realised that you meant the following.
$\displaystyle
\left(
\begin{matrix}
u & u & u\\
v & -v & 0\\
-2w & -w & w\\
\end{matrix}
\right)
$
So I formed the determinant of this matrix and got a non-zero result.
Thanks for your help.