linear independece

Hello. I have a problem here that i don't know how to start:

Determine whether this set of vectors is Linearly dependent or independent.
{u, v, w} where 4u-2v+3w = 0

Any tips on how to begin proving whehter it's indep or dependent

Last edited:

Hurkyl
Staff Emeritus
Gold Member
Do you recall the definition of linear dependence?

I know that a set is lin dependent if:
-a set contains a zero vector
-Let V1,V2,V3...Vr be vectors in Rn. If r > n, then set is dependent
-one of the vectors is a lin combo of remaining vectors in a set

Hurkyl
Staff Emeritus
Gold Member
The actual definition, as I recall, is this:

A set of vectors $x_1, x_2, \ldots, x_n$ is linearly dependent if and only if there exists scalars $a_1, a_2, \ldots, a_n$ (that are not all zero) such that $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = 0$.

Or, equivalently,

A set of vectors $x_1, x_2, \ldots, x_n$ is lienarly independent if $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = 0$ implies that each of the $a_i$ are zero.

(Not only is it the definition I recall, but one of the more useful characterizations of linear dependence!)

So the problem is fairly trivial. But it's still straightforward using the conditions you listed... for example, can you find a way to write u as a linear combination of v and w?

Those are theorems derived from the definition... the definition will give you the answer directly.

edit: too late :D

a set of vectors $${v_1, v_2,...v_k}$$ is linearly independent <==> $$a_1 v_1 + a_2 v_2 + ... + a_k v_k = 0$$ implies all the $$a_i = 0$$
that's the most basic definition of linear independence i learned. i don't really know what the problem is though. wouldn't the dimension of u, v, w matter?

I figured it out. Thank so much