We define linear independence of a set of vectors, and explore this concept algebraically and geometrically.

### VEC-0100: Linear Independence

Consider the vector equation

where are vectors of and are unknown scalar coefficients.

Can we say whether or not this equation is consistent?

Observe that this equation must be consistent, regardless of what the vectors
are, because is a solution. This solution is known as the *trivial solution*.

(Click the arrow on the right for the answer.)

*linearly independent*if the only solution to is the

*trivial solution*.

If, in addition to the trivial solution, a *non-trivial solution* (not all are zero) exists,
then we say that the set is *linearly dependent*.

If the set of vectors is linearly independent (dependent), we often say that the vectors are linearly independent (dependent).

Clearly is a solution to the equation. The question is whether another solution exists.

The vector equation translates into the following system:

Writing the system in augmented matrix form and applying elementary row operations gives us the following reduced row-echelon form: This shows that (eq:linrelationpart1) has infinitely many solutions: Letting , we obtain the following:

We conclude that the vectors are linearly dependent.

item:linindpart2 We need to solve the equation

Converting the equation to augmented matrix form and performing row reduction gives us This shows that is the only solution. Therefore the two vectors are linearly independent.

In Part item:linindpart1 of Example ex:linind we found that the vectors are linearly dependent because

Observe that (eq:nontrivrel) allows us to solve for one of the vectors and express it as a linear combination of the others. For example, This would not be possible if a nontrivial solution to the equation did not exist.

- Proof
- See Practice Problem prob:lindeplincombofother.

#### Geometry of Linearly Dependent and Linearly Independent Sets

Theorem th:lindeplincombofother gives us a convenient way of looking at linear dependence/independence geometrically. When looking at two or more vectors, we ask “can one of the vectors be written as a linear combination of the others?” If the answer is “yes”, then the vectors are linearly dependent.

##### A Set of Two Vectors

Two vectors are linearly dependent if and only if one is a scalar multiple of the other. Two nonzero linearly dependent vectors may look like this:

or like this:

Two linearly independent vectors will look like this:

##### A Set of Three Vectors

Given a set of three nonzero vectors, we have the following possibilities:

- (Linearly Dependent Vectors) The three vectors are scalar multiples of each
other.
- (Linearly Dependent Vectors) Two of the vectors are scalar multiples of each
other.
- (Linearly Dependent Vectors) One vector can be viewed as the diagonal of a
parallelogram determined by scalar multiples of the other two vectors. All three
vectors lie in the same plane.
- (Linearly Independent Vectors) A set of three vectors is linearly independent if the vectors do not lie in the same plane. For example, vectors , and are linearly independent.

### Practice Problems