Given that uv is a linearly independent subset of Rn and uvw

Given that {u,v} is a linearly independent subset of R^n, and {u,v,w} is a linearly dependent subset of R^n, show that wSpan{u,v}. (hint:write a dependence relation amont the elements of the second set, and reach a conclusion about the coefficient of w. Then finish the proof)

Solution

If u and v are linear independent subest of R^n:

c1u +c2v =0 ---> c1=c2=0

{u,v,w} is a linearly dependent subset of R^n

c3u +c4v +c5w =0

c5w = - (c3u + c4v)

w = -(c3/c5)u - (c4/c5)v

So, w is a linear comnination of u and v and can be spanned by u and v i.e can be expressed as a

combination of u and v vectors