Linear Dependence and Independence

Definition

Let be a set of vectors in vector space .

We say that is linearly dependent if there exists , not all zero, such that

We say that is linearly independent if the only solution to

is , which is called the trivial solution

(See zero vector)

Intuition

A set of vectors is linearly dependent if we can express one of the vectors as a linear combination of the others
Otherwise, it is linearly indpendent

Example - Linearly Independent

Consider the set

Let , and consider

We obtain a homogeneous system.

Carrying the coefficient matrix to REF

Since there is only one solution, this makes (since the system is homogeneous), so the set is linear independent

Example - Linearly Dependent

Consider the set

We obtain the matrix

Since there is a free variable, the set is linearly dependent

Example - Linearly Dependent

Consider the set

So the system is linearly dependent

Theorem - Linear Independence

Let be vectors in , and .

is linearly independent if and only if

That is, if the matrix is full rank, then none of the rows can reduce the other.

Theorem - Linear Dependence

A set of vectors is linearly dependent if and only if there exists an integer such that

Example

Consider the set
Then we have for any
Since there are infinite solutions, the set is linearly dependent

Example

Let be such that is linearly independent.
Prove that the set is linear independent.

For ,

Since is linearly independent, we have .
Given this, we have

Example

Let be a linearly independent set of vectors in
Prove that is linearly independent

Suppose for contradiction that is linearly dependent
Then for , there exists a ,
If we add to both sides, we have
But this means is linearly dependent, which is a contradiction.