Linear map
A linear map (also called a homomorphism of vector ) is in linear algebra is an important type of mapping between two vector spaces over the same body. In a linear mapping, it does not matter whether you add two vectors first and then maps the sum or first maps the vectors and then forms the sum of the images. The same is true for the multiplication by a scalar of the base body.
The illustrated example of a reflection in the Y-axis illustrates this. The vector is the sum of the vectors and and his image of the vector. But we also obtained when the images and the vectors and added.
One then speaks of a linear transformation with the vector addition and scalar multiplication shortcuts is compatible. This is therefore in the linear mapping is a homomorphism ( structure- preserving mapping ) between vector spaces.
In the functional analysis, when considering infinite-dimensional vector spaces which carry a topology, we speak mostly of linear operators instead of linear maps. Formally, the terms are synonymous. In infinite-dimensional vector spaces, however, the question of continuity is important, while continuity is always present at linear maps between finite dimensional real vector spaces (each with the Euclidean norm ), or more generally between finite Hausdorff topological vector spaces.
Definition
Let and be vector spaces over a common base. A mapping is called linear mapping if for all and the following conditions apply:
- Is homogeneous:
- Is additive:
The above two conditions can also be summarized as follows:
For this passes into the condition of homogeneity and in that for the additivity. Another equivalent condition is the requirement that the graph of the figure is a subspace of the sum of vector spaces and.
Examples
- For any linear map has with the shape.
- It should be and. Then, for each array with the aid of a linear transformation matrix multiplication by defined. Each linear mapping from to can be shown.
- If an open interval, the vector space of continuously differentiable functions, and the vector space of continuous functions, so is the picture, , the each function its derivative maps, linear. The same applies to other linear differential operators.
Image and core
Two important when considering linear maps quantities, the image and the kernel of a linear map.
- The image of the figure is below the amount of the image vectors, ie the set of all with out. The image set is therefore also listed by. This image is a vector subspace of.
- The core of the Figure is made of the amount of the vectors, which are represented by the vector of zero. He is a subspace of. The mapping is injective if the core contains only the zero vector.
Properties
- A linear mapping between vector spaces and forms the zero vector of starting on the zero vector by: because
- A relationship between core and image of a linear map describes the homomorphism theorem: The factor space is isomorphic to the image.
Linear maps between finite dimensional vector spaces
Base
A linear map between finite dimensional vector spaces is uniquely determined by the images of the vectors of a basis. Forming the vectors a basis of the vector space and are vectors in, then there exists a linear map, on, on, ..., on maps. Is any vector, so it can be uniquely represented as a linear combination of the basis vectors:
His image is given by
The mapping is injective if the image vectors of the basis are linearly independent. It is precisely then surjective if the target space span.
Assigns to every element of a basis of a vector of arbitrary, so you can continue with the above formula, this assignment clearly a linear mapping.
To set the field vectors with respect to a base is, this leads to the matrix representation of the linear transformation.
Mapping matrix
Those that form the entries of the matrix:
In the th column so are the coordinates of relative to the base.
Using this matrix, one can calculate the image vector of each vector:
Therefore applies to the coordinates of respect
This can be expressed by means of the matrix multiplication:
The matrix is called projection matrix or matrix representation of. Other spellings for are and.
Dimension formula
Image and core are about the dimension set in relationship. From this says that the dimension of the sum of the dimensions of the image and of the core is:
Linear mappings between infinite-dimensional vector spaces
In particular, in the functional analysis one considers linear maps between infinite dimensional vector spaces. In this context it is called the linear maps mostly linear operators. The considered vector spaces usually wear nor the additional structure of a normed complete vector space. Such vector spaces are called Banach spaces. In contrast to the finite dimensional case, it is not enough to examine only linear operators on a base. After the Baire category theorem namely has a base of infinite-dimensional Banach space uncountably many elements and are therefore no specific information for the linear operator. Also can be linear operators between infinite dimensional vector spaces not described by matrices. An exception is the class of Hilbert - Schmidt operators, these operators can be represented using "infinitely large matrices ".
Special linear maps
Vector space of linear maps
The set of linear maps from a vector space into a vector space is a vector space over, more precisely, a subspace of the vector space of all functions from to. This means that the sum of two linear mappings and component-wise defined by
Again, a linear transformation and that the product
A linear map with a scalar is also a linear map again.
, The dimension and the dimension and are placed in a base and a base, then the image in the die space
An isomorphism. Thus, the vector space has dimension.
Considering the amount of the self-images of a linear vector space, they do not only form a vector space, but to the concatenation of images as a multiplication associative algebra.
Generalization
A linear mapping is a special case of an affine transformation.
If we replace in the definition of the linear mapping between vector spaces the body through a ring, you get a Modulhomomorphismus.