A linear map is a map which maps vectors from a given space , to vectors from another linear space (), and satisfying the linear condition, which says that for every vectors and numbers we have . E.g. a rotation around is a linear map . Given two vectors and scalars we will get the same vector regardless of whether we rotate the vector first and then multiply by numbers and add them, or multiply by numbers, add and then rotate.
Therefore, to prove that a given map is a linear map we need to prove that for any two vectors and any two numbers it satisfies the linear condition. E.g. given as is a linear map because if and , then .
Meanwhile, to disprove that a map is linear we need to find an example of two vectors along with two numbers such that the linear condition fail for them. E.g. given as is not a linear map because .
Usually, linear maps will be given by their formulas. E.g. , . Then to see what does to a given vector, e.g. , we substitute it to the formula: . By the way, it is easy to see, that this is simply the rotation around by 90 degrees clockwise. Less geometrical example: let , , therefore maps vector to vector .
Sometimes we can define a linear map by giving its values on the vectors from a given basis only. This suffices to determine this map. E.g. let be given in the following form: (vectors , constitute a basis of the plane). We can calculate the formula of this map. First calculate the coefficients of the standard basis in the given one (by solving a system of equations or by guessing). In this case we see that (coefficients: and (coefficients: ). Therefore, .
In the above example we can also come to the conclusion what is and by writing down a matrix consisting of the vectors of the given basis on the left hand side and values on the right hand side. Each operation on the rows of the matrix does not change the principle that the vector is on the left, and its value on the right, because is a linear map. So after transforming the matrix to the reduced echelon form on the right hand side one will find and .
so as before
Given two linear maps and a number , their sum and are linear maps . Obviously we add and multiply by coefficients. So e.g. if , then .
It means that the set of all linear maps between fixed vector spaces and is a vector space itself. It is denoted by .
Given two linear maps and we can compose them and consider a map which transforms a vector from first via and the result via getting a vector form .
Such a map is denoted as . Given formulas defining i we can easily get the formula for . E.g. consider as above and such that . Therefore .
Simple classes of linear maps
If is a direct sum of its subspaces, then , , where , , , we call the projection onto along , and such that is the reflection across along .
The mapping , where is a subspace of and is the inclusion of into .
For , the mapping , such that is the homothety with ratio . For ratio it is the identity, denoted by . For scale it is the zero transformation.
Finally, the rotation by angle is such that
Kernel and image of a linear transformation
If is a linear transformation, then it is easy to see that
called the kernel of and
called the image of are linear subspaces of and respectively. The dimension of the image of is also called the rank of and denoted by . Notice that if spans , then spans . Moreover, if , and is a basis of , then is a basis of . Therefore,
Monomorphisms, epimorphisms and isomorphisms
A linear map which is one-to-one is called a monomorphism. It a map is ,,onto” it is called an epimorphism. If it is a bijection, we call it isomorphism.
E.g. if then the projection onto along is neither epi nor mono, but if considered as a map it is an epimorphism. The reflection across along is an isomorphism. Inclusion of a subspace into is an example of a monomorphism.
It is easy to see that a map is a monomorphism if and only if , but it is an epimorphism if .
Therefore, every monomorphism maps every linearly independent system of vectors onto a linearly independent system. Thus, every isomorphism maps every basis onto a basis. Indeed, a mapping is an isomorphis if and only if it maps a basis onto a basis. Therefore, if is a isomorphism, then . Moreover, if is finite and , then the notions of isomorphism, monomorphism and epimorphism coincide.
Two spaces are isomorphic if there is an isomorphism between them, and we denote this by . Notice that two finitely dimensional subspaces over a field are isomorphic if and only if the dimensions are equal (to see that it is true, consider bases!).
It is easy to prove that is
- an epimorphism if and only if there exists a mapping such that ,
- a monomorphism if and only if there exists a mapping such that ,
- an isomorphism if and only if there exists a mapping such that and .
In the last case is called the inverse map to and denoted by .
E.g. if and , then .