Basis (linear algebra) — Safekipedia Discoverer

In mathematics, a special group of numbers or points called a basis helps us describe every point in a space using simple building blocks. Imagine you have a puzzle where each piece can be mixed in just the right way to create any picture. The pieces you use are like the basis for that picture. In math, these building blocks are called basis vectors.

A basis must have two important properties. First, the basis vectors must be linearly independent, meaning no single vector can be made by mixing the others. Second, every point in the space should be able to be made by adding up multiples of these basis vectors, which we call a linear combination. When we do this, the numbers we multiply each basis vector by are called components or coordinates.

One amazing thing about bases is that no matter which basis you pick for a space, they all have the same number of vectors. This number is called the dimension of the space. For example, in regular 3D space, we use three basis vectors to describe any point, which tells us the space is three-dimensional.

Basis vectors are useful in many areas, like studying how crystals are structured or setting up different frames of reference to look at objects from new angles. Understanding bases helps us see how complex spaces are built from simple parts.

Definition

A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a special set of vectors that can be used to describe every vector in the space uniquely.

To be a basis, the set must have two key properties: it must be linearly independent, meaning no vector in the set can be made by combining the others, and it must span the space, meaning any vector in the space can be created by combining vectors from the set. When these conditions are met, every vector in the space can be written in exactly one way using the basis vectors.

Examples

The set R² of ordered pairs of real numbers is a vector space. A simple basis for this space consists of the two vectors e₁ = (1, 0) and e₂ = (0, 1). These vectors are called the standard basis because any vector v = (a, b) in R² can be written uniquely as v = ae₁ + be₂.

More generally, if F is a field, the set Fⁿ of n‑tuples of elements of F is also a vector space. The standard basis for Fⁿ consists of vectors where each has a 1 in one position and 0s elsewhere. For example, in F³, the basis vectors are (1, 0, 0), (0, 1, 0), and (0, 0, 1).

Another example comes from polynomial rings. If F is a field, the collection F[X] of all polynomials in one indeterminate X with coefficients in F is a vector space. One basis for this space is the monomial basis, which includes all monomials such as 1, X, X², and so on. Any set of polynomials that includes exactly one polynomial of each degree can also serve as a basis.

Properties

In linear algebra, a basis is a special set of vectors that can be combined in unique ways to make every vector in a vector space. The Steinitz exchange lemma helps us understand how to replace some vectors in a larger set with others to still cover the whole space.

For a vector space of dimension n, a set of n vectors will be a basis if it is either linearly independent (meaning none of the vectors can be made from combining the others) or a spanning set (meaning they can be used to create every vector in the space). All bases for the same space have the same number of vectors, which is called the dimension of the space.

Coordinates

Imagine you have a special set of building blocks called a basis. These blocks help you build any object in a certain space, and each object can only be built in one special way using these blocks. The numbers you need to use for each block to build an object are called coordinates.

When we list the blocks in a specific order, we can easily find the coordinates for any object. This ordered list of blocks helps us understand the position of objects in space, just like how addresses help us find places on a map.

Change of basis

Main article: Change of basis

When we have two different ways to describe the same set of points in space, called bases, we can change from one to the other. This helps us see the same information in a new way.

For example, if we know how to write points using one basis, we can use a special rule to write them using another basis. This rule uses a matrix, which is like a grid of numbers, to switch between the old and new ways of describing points.

Related notions

Free module

Main articles: Free module and Free abelian group

When we think about vectors but use numbers that are not just whole numbers, we get something called a module. Like vectors, we can find special sets of numbers in a module that help build all other numbers in that module. These special sets are called bases, just like in vectors. However, not all modules have these special sets. When a module does have one, it is called a free module.

Analysis

Empirical distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n-dimensional cube [−1, 1]n as a function of dimension, n. Boxplots show the second and third quartiles of this data for each n, red bars correspond to the medians, and blue stars indicate means. Red curve shows theoretical bound given by Eq. (1) and green curve shows a refined estimate.

When we work with very large spaces of vectors, we sometimes need different kinds of bases. One type is called a Hamel basis, which works like the bases we learned about before but can become very large and hard to use. Other types of bases, like orthogonal bases, let us combine infinitely many vectors in special ways to build all the vectors in the space. These other types are often easier to work with in advanced math.

Geometry

In geometry, there are special sets of points that act like bases in different spaces. For example, in an affine space (a space where we can add and subtract points), we need just a few points to describe the whole space. Similar ideas help us understand shapes and positions in higher dimensions.

Random basis

Sometimes, we can pick vectors at random and they will form a basis almost always. This helps scientists and mathematicians when they need to approximate bases without checking every possibility carefully.

Proof that every vector space has a basis

Every vector space has a basis, which is a special set of vectors that can be combined to make any other vector in the space. To show this, we look at all sets of vectors that don’t rely on each other and use a mathematical rule called Zorn's lemma. This helps us find the biggest such set, which ends up being a basis because it can create every vector in the space while still not relying on each other.

This idea connects closely to something called the axiom of choice, meaning that if we can always find a basis for any vector space, then the axiom of choice must be true, and vice versa.