Question

proof for if two vectors ar linearly dependent that one is a scalar multiple of the other

Answer 1

Suppose two vectors u and v are linearly dependent.
If either u or v is the zero vector, say u = 0, then u = 0*v.

If neither u nor v are zero vectors,
then there exists scalars a1 and a2, not both zero, such that
a1*u + a2*v = 0.

If a1 is 0, then a2 * v = 0, implying a2 = 0
(since v is nonzero). a1 = a2 = 0 contradicts our assumption
that we don’t have both scalars being zero.

Therefore, a1 can’t be zero.
Similarly, a2 can’t be zero.
And so they’re both nonzero scalars.

Then from a1*u + a2*v = 0,
solving for u, we get u = -(a2/a1) * v.
So u is a scalar multiple of v.