Linear independence

In linear algebra, we learn about linear independence. This helps us understand how vectors — like arrows pointing in space — relate to each other.

A group of vectors is linearly independent if none of the vectors can be made by mixing the others together in a special way. We call this mixing a linear combination.

Linearly independent vectors in R 3 {\displaystyle \mathbb {R} ^{3}}

If one vector can be made by mixing the others, then the group is linearly dependent. Knowing if vectors are independent or dependent helps us understand the space they are in. It’s important when we study something called a linear basis, which is the smallest set of vectors we need to describe everything in that space.

Figuring out linear independence helps us measure how big or complex a vector space is. For more on how this idea connects to statistics, you can look at [Independence (statistics)](/wiki/Independence_(statistics) and Covariance.

Definition

A group of vectors is linearly dependent if one vector can be made by adding together the other vectors, using numbers called scalars. This means that one vector is not special — it is just a mix of the others.

If the vectors are linearly independent, then no vector can be made from the others. Each vector is unique and important on its own. This idea helps us understand how vectors work together in spaces.

Main article: Linear independence

Geometric examples

Two arrows, u and v, pointing in different directions on a flat surface are linearly independent. This means you can't describe one arrow using just the other. But if you have three arrows all lying flat on the same surface, they are linearly dependent because one arrow can be described using the other two.

An arrow pointing straight up (k) is not flat like u and v, so u, v, and k together are linearly independent. They help describe points not just on the flat surface, but also in three dimensions.

Geographic location

Imagine telling a friend a place is "3 miles north and 4 miles east" from where they stand. These two directions — north and east — are linearly independent because you can't describe north using only east, or east using only north. If you also say the place is "5 miles northeast," this is just a mix of the north and east directions, so it doesn’t give new information. To describe places in three dimensions, like adding height, you would need a third direction, such as up or down.

plane linear combination

Evaluating linear independence

Linear independence is an important idea in linear algebra. A group of vectors is linearly independent if no vector in the group can be made by adding together other vectors in the group, multiplied by numbers (called scalars).

The zero vector is special. If a group of vectors includes the zero vector, the group is always linearly dependent. This is because the zero vector can be made by using just itself—multiply it by 1 and all other vectors by 0.

With two vectors, they are linearly dependent if one is just a number times the other. This means you can multiply one vector by a number to get the other vector. If neither vector can be made this way from the other, then they are linearly independent. This idea can be used with more vectors, using methods like row reduction or determinants.

Natural basis vectors

Let ( V = \mathbb{R}^n ) and think about special vectors in ( V ) called the natural basis vectors. These vectors look like this:

( \mathbf{e}_1 = (1, 0, 0, \ldots, 0) )
( \mathbf{e}_2 = (0, 1, 0, \ldots, 0) )
...
( \mathbf{e}_n = (0, 0, 0, \ldots, 1) )

These vectors are linearly independent. This means you cannot create any one of these vectors by adding together multiples of the other vectors. This idea is important because it helps us understand how vectors can combine to form other vectors in space.

Linear independence of functions

Let’s learn about linear independence using functions. Imagine we have a group of functions, like recipes, that we can mix together.

Two functions are linearly independent if we can’t make one by mixing the other using just numbers and addition.

For example, consider the functions ( e^t ) and ( e^{2t} ). These are special patterns that change over time. To show they are linearly independent, we check if there’s any way to combine them to get zero everywhere. We find that the only way this works is if both numbers used in the mix are zero. This proves that ( e^t ) and ( e^{2t} ) are linearly independent — you can’t create one from the other!

Space of linear dependencies

A linear dependency among vectors v₁, ..., v_n is a way to mix these vectors using numbers (a₁, ..., a_n) so that the total equals zero. If at least one of these numbers is not zero, the vectors are called linearly dependent.

Linear dependencies among vectors create their own vector space. When we look at vectors using their coordinates, finding these dependencies is like solving a special puzzle. This can help us find a simple set of dependencies using a method called Gaussian elimination.

Generalizations

A set of vectors is affinely dependent if one vector can be made by mixing the others in a special way, called an affine combination. If this is not possible, the set is affinely independent. Every affinely dependent set is also linearly dependent, meaning that linearly independent sets are always affinely independent too.

Two vector spaces are linearly independent if they share only the zero vector. This idea can be extended to many vector spaces at once, where each space shares only the zero vector with the combination of all the others. When this happens, the big space is called a direct sum of the smaller spaces.