Linear independence — Safekipedia Discoverer

In linear algebra, we talk about something called linear independence. This is a way to describe how vectors — which are 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 whether vectors are independent or dependent helps us understand the space they live in. It’s a key idea when we study something called a linear basis, which is like the smallest set of vectors we need to describe everything in that space.

Whether a space has a limited number of independent vectors or an endless number depends on its size. Figuring out linear independence helps us measure how big or complex a vector space really is. For more on how this idea connects to statistics, you can look at Independence (statistics) and Covariance.

Definition

A group of vectors is called linearly dependent if one of the vectors can be made by combining the others using numbers called scalars. This means that one vector isn’t unique — it’s just a mix of the others.

If the vectors are linearly independent, then no vector can be made by mixing the others. Each vector stands on its own, and you can’t create any of them from the rest of the group. This idea helps us understand how vectors relate to each other in spaces.

Main article: Linear independence

Geometric examples

Two arrows, u and v, that point in different directions on a flat surface are linearly independent. This means you can't describe one arrow using just the other arrow. However, if you have three arrows all lying flat on the same surface, they are linearly dependent because one of them 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 combination 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 a key concept in linear algebra. A set of vectors is linearly independent if no vector in the set can be written as a combination of the others. If such a combination exists, the vectors are linearly dependent.

The zero vector plays a special role. If a set of vectors includes the zero vector, the set is automatically linearly dependent. This is because the zero vector can always be expressed as a trivial combination of other vectors. For example, if one vector is the zero vector, we can set its coefficient to 1 and all other coefficients to 0, resulting in a valid linear combination that equals the zero vector.

When dealing with two vectors, they are linearly dependent if one is a scalar multiple of the other. This means that one vector can be scaled (multiplied by a number) to become the other. If neither vector is a scalar multiple of the other, then they are linearly independent. This concept extends to higher dimensions, where more complex methods, such as row reduction or determinants, can be used to determine linear independence.

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, meaning you can't make any one of them by adding up multiples of the others. This is important because it helps us understand how vectors can build up other vectors in space.

Linear independence of functions

Let’s explore a special idea called linear independence using functions. Imagine we have a group of functions, like recipes, that can be mixed together in specific ways. Two functions are linearly independent if you 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 (by multiplying by numbers and adding) to get zero everywhere. Through some math steps, 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 set of numbers (a₁, ..., a_n) that shows how these vectors can combine to equal zero. When such a combination exists with at least one number not being zero, the vectors are called linearly dependent.

Linear dependencies among vectors form their own vector space. When vectors are shown by their coordinates, finding these dependencies is like solving a special kind of equation system. This helps in finding a basic 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.