# Thread: Another linear dependence problem

1. ## Another linear dependence problem

if $\{u,v,w,\}$ is a linearly independent set of vectors, then $S = \{u+v-2w,u-v-w,u+w\}$ is also linearly independent.

True or false?

thanks

2. Originally Posted by Bucephalus
if $\{u,v,w,\}$ is a linearly independent set of vectors, then $S = \{u+v-2w,u-v-w,u+w\}$ is also linearly independent.

True or false?

thanks
Form the determinant of those coefficients and show it is non-zero.

3. Originally Posted by ThePerfectHacker
Form the determinant of those coefficients and show it is non-zero.
At first I set up the matrix with each vector in a row like this:

$
\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.

$
\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.