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 when you add two vectors together and then apply the linear map, it’s the same as applying the map to each vector first and then adding the results. Similarly, multiplying a vector by a number before or after applying the map gives the same outcome.

One common way to represent a linear map is with an m × n matrix. This matrix takes vectors from a space with n dimensions and turns them into vectors in a space with m dimensions, following the rules of linear maps. For example, a linear map always sends the origin—the point where all coordinates are zero—in the starting space to the origin in the ending space. It also sends flat surfaces that pass through the origin, like planes or lines, to other flat surfaces in the ending space, which might be planes, lines, or just the single origin point.

Linear maps are important because they appear in many areas of mathematics and science. Simple examples include rotation and reflection linear transformations, which are used to change the orientation 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 tools for solving many kinds of problems.

Definition and first consequences

A linear map is a special kind of function used in math, particularly in a subject called linear algebra. It connects two spaces called vector spaces. These spaces have objects known as vectors, which 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 apply the function, it should give the same result as applying the function to each vector separately and then adding the results. Second, if you multiply a vector by a number and then apply the function, it should be the same as applying the function to the vector first and then multiplying the result by that number. These rules mean the function preserves the operations of addition and scalar multiplication.

This idea can also apply to more complex situations with many vectors and numbers combined, and it always preserves combinations of vectors in a consistent way.

Examples

Linear maps are functions between vector spaces that preserve vector addition and scalar multiplication. A simple example is multiplying each component of a vector by a constant. For instance, the function that turns every vector (x, y) into (2x, y) is linear because it doubles the x-component while leaving the y-component unchanged.

Another example is the zero map, which sends every vector to the zero vector. This also qualifies as a linear map because it respects both vector addition and scalar multiplication. Matrices are also linear maps; they take vectors as input and produce new vectors through multiplication, maintaining the properties of linearity.

Matrices

Main article: Transformation matrix

In math, especially in linear algebra, a linear map is a special kind of function between spaces of vectors. It follows simple rules for adding vectors and multiplying them by numbers.

When we have spaces with a fixed number of dimensions and we pick a basic set of vectors (called a basis), we can show a linear map using a matrix. This helps us do calculations more easily. For example, a matrix with rows and columns can change a vector in one space to a vector in another space, following the same addition and multiplication rules.

Vector space of linear maps

Linear maps can be combined in special ways. If you have two linear maps, you can perform an operation called "composition" where you apply one map, and then apply the second map to the result. This new combined map is also linear.

The set of all linear maps between two spaces forms a vector space itself. This means you can add linear maps together or multiply them by numbers, and the results stay linear. When the starting and ending spaces are the same, these maps form a special structure called an associative algebra, where the order of composition does not affect the final result.

Kernel, image and the rank–nullity theorem

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

In linear algebra, when we have a special kind of function called a linear map, 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. Both of these sets have special properties and can be measured by how many separate directions or "dimensions" they contain.

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

Cokernel

Main article: Cokernel

In linear algebra, a co-kernel is a way to understand how a linear map behaves. It is linked to the kernel, which tells us about solutions to certain equations. While the kernel is a space inside the starting area, the co-kernel is about the space outside the ending area.

Think of it like solving a puzzle: the kernel tells us what pieces fit perfectly, while the co-kernel tells us what pieces are missing. This helps us understand 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 has different classifications based on its properties.

One type is called injective or a monomorphism, meaning it never maps two different vectors to the same result. Another type is surjective or an epimorphism, meaning every vector in the output space is 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, which is important in many areas of mathematics.

Continuity

Main articles: Continuous linear operator and Discontinuous linear map

In mathematics, a linear transformation between special types of vector spaces, called topological vector spaces, can either be continuous or not. When the space where the transformation starts and ends are the same, it is called a continuous linear operator. For example, in spaces with limits, a linear operator is continuous if it stays within certain limits. However, in more complex spaces, there can be linear transformations that are not continuous. One example is taking the rate of change of smooth functions, which can sometimes produce very large results even from small inputs.

Applications

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

Linear maps are also important in making computer programs run better. They help 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.