This is an exercise from Jänich's *Linear Algebra* (exercise 3.1). Let $V$ be a real vector space and $a, b, c, d \in V$. Suppose that:

\begin{align\*} v\_1 &= a&&+b&&+c&&+d\\
v\_2 &= 2a&&+2b&&+c&&-d\\
v\_3 &= a&&+b&&+3c &&-d\\
v\_4 &= a&&{}&&-c &&+ d\\
v\_5 &= &&-b&&+c&&-d
\end{align\*}

The exercise is to show that $(v\_1, \dots, v\_5)$ are linearly dependent. Now, it's not very difficult to show, e.g., that $v\_5$ can be expressed as a linear combination of the other four vectors. But Jänich also says that this exercise can be solved without doing "any calculations", so I'm curious if there is some other criterion for linear dependence that would immediately apply to this exercise.

My hunch is that if $v\_1, \dots, v\_n$ (all pairwise distinct) can each be expressed as the linear combination of $w\_1, \dots, w\_m$ vectors for $m \leq n$, then $(v\_1, \dots, v\_n)$ is linearly dependent. I thought about proving this by induction on $m$, but couldn't get very far with the induction step. Is this in the right direction? Or am I totally missing the point of Jänich's remark (and if so, could some one give me a hint as to what he is getting at)?