Let $a$, $b$ and $c$ be a system of non-zero, non-coplanar vectors. Find a relation between the vectors $f$, $g$, $h$, $j$ given by:
$$f=a+3b+4c$$
$$g=a-2b+3c$$
$$h=a+5b-2c$$
$$j=6a+14b+4c$$
How can we show this?
My attempt:
Let's assume that they are linearly independent.
$$αf+βg+γh+λj=0 -eq(1)$$
Where each of the scalars is equal to zero $α = β = γ = λ =0$.
Substituting $f$, $g$, $h$, $j$ into (1) we get:
$$α + β + γ + 6λ=0$$ $$3α -2β + 5γ + 14λ=0$$ $$4α + 3β - 2γ + 4λ=0$$
Using this result, how do I prove that at least one of these scalars is zero or at least one of them is non-zero?
Here is an example way the textbook solves this: Finding the whether vectors are linearly independent or dependent