Basis (linear algebra) — Safekipedia Adventurer

In mathematics, a basis is a special group of numbers or points that helps us describe every point in a space using simple building blocks. Think of it like having puzzle pieces that can be mixed just right 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 rules. First, the basis vectors must be linearly independent. This means 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. We call this a linear combination. The numbers we multiply each basis vector by are called components or coordinates.

One cool 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. This 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 group of vectors. These vectors help describe every vector in the space in one clear way.

To be a basis, the group must have two important rules: it must be linearly independent, meaning no vector can be made from mixing the others. It must also span the space, meaning any vector can be made by mixing vectors from the group. When both rules are true, every vector can be written in just one way using the basis vectors.

Examples

The set R² of ordered pairs of real numbers is a special kind of space called a vector space. A simple basis for this space has 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 as v = ae₁ + be₂.

More generally, if F is a field, the set Fⁿ of groups of n elements from F is also a vector space. The standard basis for Fⁿ has vectors with a 1 in one place and 0s everywhere else. 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 has one polynomial of each degree can also be a basis.

Properties

In linear algebra, a basis is a special set of vectors. These vectors can be mixed in unique ways to make every vector in a space.

The Steinitz exchange lemma helps us understand how to replace some vectors in a larger set with others and still cover the whole space.

For a 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. This number 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. Each object can only be built in one special way using these blocks. The numbers you need 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. Modules are like vectors, and they can have special sets of numbers that help build all other numbers in the 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. A basis is a special set of vectors. We can combine them to make any other vector in the space.

To show this, we look at sets of vectors that do not rely on each other. We use a mathematical rule called Zorn's lemma. This helps us find the biggest such set. This set is a basis. It can create every vector in the space and still does not rely on each other.

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