Linear map

In mathematics, especially in linear algebra, a linear map is a special kind of function that works between vector spaces. It follows two important rules: it respects vector addition and scalar multiplication. This means that adding two vectors together and then using the linear map gives the same result as using the map on each vector first and then adding the results. Multiplying a vector by a number before or after using the map also gives the same outcome.

One common way to show a linear map is with an m × n matrix. This matrix changes vectors from a space with n dimensions into vectors in a space with m dimensions, following the rules of linear maps. For example, a linear map always moves the origin—the point where all coordinates are zero—in the starting space to the origin in the ending space. It also changes flat surfaces that pass through the origin, like planes or lines, into other flat surfaces in the ending space.

Linear maps are important because they are used in many areas of mathematics and science. Simple examples include rotation and reflection linear transformations, which change the direction of shapes in geometry. Because linear maps can often be shown as matrices, they are easy to work with using algebra and computers, making them useful for solving many kinds of problems.

Definition and first consequences

A linear map is a special kind of function used in math, especially in a subject called linear algebra. It connects two spaces called vector spaces. These spaces have objects known as vectors. Vectors can be added together or multiplied by numbers (called scalars).

For a function to be a linear map, it must follow two important rules. First, if you add two vectors together and then use the function, it should give the same result as using the function on each vector separately and then adding the results. Second, if you multiply a vector by a number and then use the function, it should be the same as using the function on the vector first and then multiplying the result by that number. These rules mean the function keeps the operations of addition and scalar multiplication working correctly.

This idea can also work with more complex situations with many vectors and numbers combined, and it always keeps combinations of vectors consistent.

Examples

Linear maps are special rules that work with points in space. They keep the space organized in a certain way.

One easy example is multiplying each part of a point by a number. For instance, the rule that turns every point (x, y) into (2x, y) is linear because it only changes the x-part, making it twice as big, while the y-part stays the same.

Another example is the zero map. This rule sends every point to the point (0, 0). It is also linear because it follows the same organized rules.

Matrices are also linear maps. They take points as input and create new points by multiplying, while keeping the organized rules of linear maps.

Matrices

Main article: Transformation matrix

In math, a linear map is a special kind of function that works with vectors. It has simple rules for adding vectors and multiplying them by numbers.

When we have spaces with a fixed number of dimensions, we can use a matrix to show a linear map. This makes calculations easier. A matrix can change a vector in one space to a vector in another space, using the same rules for addition and multiplication.

Vector space of linear maps

Linear maps can be combined in special ways. If you have two linear maps, you can do something called "composition." This means you use one map, and then use the second map on the result. The new map you get is also linear.

All the linear maps between two spaces form a vector space. This means you can add linear maps together or multiply them by numbers, and they will still be linear. When the starting and ending spaces are the same, these maps form a special structure called an associative algebra. In this structure, the order in which you combine the maps does not change the result.

Kernel, image and the rank–nullity theorem

Main articles: Kernel (linear algebra), Image (mathematics), and Rank of a matrix

In linear algebra, a linear map is a special kind of function. We can study two important parts of it: the kernel and the image.

The kernel is the set of all inputs that the map sends to zero. The image is the set of all outputs that the map can produce.

A key idea is that the total number of directions in the input space equals the sum of the directions in the kernel and the image. This is called the rank–nullity theorem.

Cokernel

Main article: Cokernel

In linear algebra, a co-kernel helps us understand how a linear map works. It is connected to the kernel, which tells us about solutions to some equations. While the kernel is a space inside the starting area, the co-kernel is about the space outside the ending area.

Imagine solving a puzzle: the kernel shows us which pieces fit just right, while the co-kernel shows us which pieces are missing. This helps us know how many answers we can find and what rules those answers must follow.

Algebraic classifications of linear transformations

A linear map is a special kind of function in math that works with vectors. It can be grouped in different ways based on its properties.

One type is called injective or a monomorphism. This means it never sends two different vectors to the same place.

Another type is surjective or an epimorphism. This means every vector in the output space can be reached by the map.

When a linear map is both injective and surjective, it is called an isomorphism.

Change of basis

Main articles: Basis (linear algebra) and Change of basis

When we look at a linear map, we can think of it like a rule that changes one set of directions into another. If we start with a set of directions called basis B, and we want to see how the linear map looks using these new directions, we use a special formula. This formula helps us find the new version of the linear map, called A′, by using the original map A and the directions in basis B.

This shows how linear maps behave when we switch from one set of directions to another. This is important in many areas of mathematics.

Continuity

Main articles: Continuous linear operator and Discontinuous linear map

In mathematics, a special kind of math rule called a linear transformation can work in two ways between vector spaces. These spaces are special because they follow certain math rules.

When the transformation starts and ends in the same space, it is called a continuous linear operator. This means the transformation works smoothly without sudden jumps.

Sometimes, in more complicated spaces, there can be linear transformations that do not work smoothly. These are called discontinuous linear maps.

Applications

Linear maps are very useful in many areas. In computer graphics, they help change the shape and position of objects, like turning or resizing pictures. These changes use special tables called transformation matrices.

Linear maps also help make computer programs run better. They improve how code with repeating steps works and can make programs run faster on many computers at once, using compiler optimizations and parallelizing compiler techniques.