Thread: Span and linearly independence question

Suppose that {u; v} is a linearly independent set of vectors and w is not in the span of {u; v}. Show that the set {u, v,w} is linearly independent.

So we no that a linearly independent set:
x1v1+x2v2+.......xpvp=0 where x has only the trivial soltuion
So does that mean i have to plug in u and w into that equation to show it will equal 0?
Im confused on how to show it when w is not in the span. Any Help would be appriciated

Originally Posted by kensington

Suppose that {u; v} is a linearly independent set of vectors and w is not in the span of {u; v}. Show that the set {u, v,w} is linearly independent.

So we no that a linearly independent set:
x1v1+x2v2+.......xpvp=0 where x has only the trivial soltuion
So does that mean i have to plug in u and w into that equation to show it will equal 0?
Im confused on how to show it when w is not in the span. Any Help would be appriciated

Supose . If then

dividing by we'd get that w is a lin. comb. of u,v, so it must be we're done.

Tonio

in a different way,
Let be scalars such that
if we had we would have which implies and that's not true by assumption.Therefore .As a result, we go back to .And.. we're done.