I'm practicing my proof-writing and was hoping you could let me know if this proof looks good. I would like to know if the proof is incorrect, if there are parts that are overly wordy/complicated, or if I'm missing some element of a proof that is helpful to see, if not strictly necessary.
The Prompt (from here):
Two nonzero vectors are linearly independent if and only if they are not colinear (or proportional, i.e. for two vectors (u,v), there exists λ ∈ R such that u = λ.v)
Sufficiency: Suppose two nonzero vectors v1 and v2 are linearly independent, but let them be colinear so that v1 = λv2. Then:
t1v1 + t2v2 = 0
λt1v2 + t2v2 = 0
v2(λt1 + t2) = 0
This implies that λt1 = -t2 even for nonzero values of t1 and t2, which contradicts the fact that v1 and v2 are linearly independent, so v1 and v2 must not be colinear.
Necessity: We prove the necessary condition by the contrapositive. Suppose v1 and v2 are not linearly independent, so that there is some nonzero t ∈ R2 such that t1v1 + t2v2=0. Then t1v1 = -t2v2 so v1 = -(t2/t1)v2 so v1 and v2 are proportional (by the scalar -t2/t1) and colinear. Since v1 and v2 not being linearly independent implies them being colinear, if they are not colinear then they are linearly independent.