Will you please check this proof? It feels like I'm cheating.
Prove that two vectors, a and b are linearly dependent if and only if a is a scalar multiple of b.
If a = kb, then a - kb = 0, and the system is linearly dependent.
If, however, a ≠ kb, then a - kb ≠ 0, and the system is not linearly dependent.
My book gives the hint that we should consider separately the case where a = 0 (the zero vector), but that just seems superfluous and unnecessary to me.
What do you guys think?
El que pega primero pega dos veces.
What definition of linearly dependant have you been given? As far as I can remember, a and b are linearly dependant iff there exists scalars a and b such that aa+bb = 0.
The first part of your proof correctly shows that a being a scalar multiple of b implies that such scalars exist, and so a and b are linearly dependant. The second part of your proof, I think, can be made more rigorous by saying that a is not a scalar multiple of b implies that there exists no k such that a=kb, so the scalars required for linear dependance do not exist.
All in all though, I think your proof is valid.
Bad speling makes me [sic]