So we have a statement

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

We assume that u and v are linearly dependant. So by definition there exits scalers such that and Now if we solve this equation for u we get . Therefore u and v are multiples of each other. Done.

Now for the other direction

We assume that there are mulitples of each other so we get

where Now we subract v from both sides to get

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.