• EvaBugs
In summary, the set of vectors {v_1, v_2,...v_k} 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 + ... + a_n x_n = 0.
EvaBugs
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:
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

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

## What is linear dependence?

Linear dependence refers to a relationship between two or more vectors in which one vector can be expressed as a linear combination of the others. In other words, one vector is a multiple of another, or a combination of the others.

## What is linear independence?

Linear independence refers to a relationship between two or more vectors in which none of the vectors can be expressed as a linear combination of the others. In other words, the vectors are not dependent on each other and cannot be written as multiples or combinations of each other.

## What is the difference between linear dependence and linear independence?

The main difference between linear dependence and linear independence is the relationship between the vectors. Linear dependence means that one vector is a combination of the others, while linear independence means that the vectors are not related in this way and cannot be expressed as combinations of each other.

## Why is linear independence important?

Linear independence is important because it allows us to determine the dimension and basis of a vector space. It also helps us to identify the number of parameters needed to describe a system, and to solve systems of linear equations.

## How do you test for linear dependence or independence?

To test for linear dependence or independence, we can use methods such as Gaussian elimination, finding the determinant of a matrix, or using the rank-nullity theorem. These methods help us to determine if a set of vectors is linearly dependent or independent.

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