The question:

Let $v_1, v_2$, and $v_3$ be distinct elements of a vector space $V$ over a field $F$ and let $c_1, c_2, c_3 \in F$. Under what conditions is the subset $G=\{c_2v_3 - c_3v_2,\, c_1v_2 - c_2v_1, c_3v_1 - c_1v_3\}$ of $V$ linearly dependent?

What I know:

1) If a set is linearly dependent, then at least one of the vectors can be written as a linear combination of the others.

2) In order for $G$ to be linearly dependent, there must exist scalars $a,b,c\in F$, not all of which are zero, satisfying

$$a(c_2v_3 - c_3v_2)+b(c_1v_2 - c_2v_1)+c(c_3v_1 - c_1v_3)=0$$

3) If $a=b=c=1$ and if $c_1=c_2=c_3$, the condition in 2) is satisfied.

This is a homework problem from a graduate level Linear Algebra course. I don't need the answer, but some direction would be great.


The simple case is when $\{v_1,v_2,v_3\}$ is a linearly dependent set. In this case the span of the set has dimension less than or equal to $3$, so any set with three elements is linearly dependent.

Suppose instead $\{v_1,v_2,v_3\}$ is linearly independent and let $U$ be the span. Then the set is a basis of this vector space and the coordinates of the new vectors with respect to it form the matrix $$ \begin{bmatrix} 0 & -c_2 & c_3 \\ -c_3 & c_1 & 0 \\ c_2 & 0 & -c_1 \end{bmatrix} $$ whose determinant is $0$.

Therefore the set $G$ is linearly dependent for all values of $c_1$, $c_2$ and $c_3$.


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