Let u,v be distinct vectors in a vector space V. Show that {u,v} is linearly
dependent if and only if u or v is a multiple of the other.
So we have a statement
$\displaystyle p \iff q$
where p is if u,v are linearly dependent and q is if u or v is a multiple of the other
Since we are proving and if and only if statement we need to prove both directions. So lets start with
$\displaystyle p \implies q$
We assume that u and v are linearly dependant. So by definition there exits scalers $\displaystyle c_1,c_2 \in \mathbb{R}$ such that $\displaystyle c_1,c_2 \ne 0$ and $\displaystyle c_1u+c_2v=0$ Now if we solve this equation for u we get $\displaystyle u=-\frac{c_2}{c_1}v$. Therefore u and v are multiples of each other. Done.
Now for the other direction
$\displaystyle q \implies p$
We assume that there are mulitples of each other so we get
$\displaystyle v=ku$ where $\displaystyle k \ne 0$Now we subract v from both sides to get
$\displaystyle 0=ku-1\cdot v$ Now we have to non zero scalers and a linear combination that is equal to zero. So the vectors are dependant.
QED.
I hope this helps.