wouldn't that be circular logic?
what I mean is
-solving linear independence is an earlier topic
-solving systems of linear equations is a later topic
But to do linear independence requires solving systems of linear equations which we haven't covered yet.
For example, determine if the set
$$ S = {\{(1, −1, 2),(1, −2, 1),(1, 1, 4)}\} $$
of three vectors in $\Bbb R^3$ is independent or dependent.
Is there a nontrivial solution to the vector equation x(1, −1, 2) + y(1, −2, 1) + z(1, 1, 4) = (0, 0, 0)?
Solve the system of homogeneous linear equations
x + y + z = 0
−x −2y + z = 0
2x + y + 4z = 0
I hope I am making sense.
Linear independence of vectors is a basic concept defined independently in ealier topic and which requires, in general, the solution of a homogeneous system of linear equation to be checked.
There is a link between the two concept but there is not any circular logic in that.