Check whether a system {$v_1,...,v_m$} of vectors in $\mathbb R^n$ (in $\mathbb R[x]$) is linearly independent.
These are my thoughts:
For {$v_1,...,v_m$} to be linearly independent, prove that:
$\lambda_1v_1 + \lambda_2v_2 + ... + \lambda_mv_m = \theta$ where $\lambda_1, ..., \lambda_m \in F$ and $\lambda_1 +... + \lambda_m = 0$ and $\theta$ represents the null vector ($\underline 0$).
So I'm assuming that each vector is in $\mathbb R^n$ but there are $m$ vectors in this system so I wrote them as a linear combination with scalars. Is this right?
So now how is the best way to do this proof. Can you do it in $\mathbb R$ and $\mathbb R^2$ then generalize or is that normally not allowed?
Will you need to use mathematical induction to generalize? I would just like some advice before I waste my time.