Suppose that S u v is a set of two vectors in a vector spac

Suppose that S = {u, v} is a set of two vectors in a vector space V. Prove that S is linearly dependent if and only if u is a scalar multiple of v or v is a scalar multiple of u.

Solution

S= {u,v} is a set of two vectors in a vector space of V.

if u and v are in vector space V

we can write it in linear combination of vectors

V= a1 u +a2 v (a1,a2 are constant)

to S is a set of u,v vectors so

S= p u +q v (here p,q are any numbers)

to be linearly dependent

pu +qv=0

pu =-qv

so S is linearly dependent if and only if u is a scalar multiple of v or v is scalar multiple of u

hence proved