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
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.