A linear map (or linear transformation) between two finite-dimensional vector spaces can always be represented by a matrix, called the matrix of the linear map. If we apply the map to an element of the first vector space, then we obtain a transformed element in the second space. Similarly, when we multiply the matrix of the map by the coordinate vector of the starting element, we obtain the coordinate vector of the transformed element.

In order to fully understand this lecture, we need to remember two things.

First, given two vector spaces and , a function is said to be a linear map if and only iffor any two vectors and any two scalars and .

Second, given a basis for and a vector , the coordinate vector of is the vector that contains the unique set of coefficients that appear in the representation of as a linear combination of the basis:

We are going to write the coordinate vector of as

Note that the concept of a coordinate vector is well-defined only if the basis of has a finite number of elements, that is, if the dimension of is finite. We are always going to assume that this is the case.

We are now ready to define the matrix of a linear map.

Definition Let and be two vector spaces. Let be a basis for and a basis for . For any and any , denote by and their and coordinate vectors with respect to the two bases of and respectively. Let be a linear map. An matrix such that, for any ,is called a matrix of the linear map with respect to the bases and .

Although it should already be clear, we highlight that the transformed vectors belong to and their coordinate vectors are vectors. Moreover, is the matrix product of and .

Example Consider the space of first-order polynomials of the formand the space of second-order polynomials of the formFor brevity, we are often going to denote a polynomial by , omitting the argument . We already know that a space of polynomials is a vector space. Moreover, if we define then is a basis for and is a basis for . The coordinate vectors of and above with respect to these two bases areLet be the map that transforms any polynomial into another polynomial equal to , that is, Take two scalars and , the polynomial defined above and another polynomial defined asThen,As a consequence is a linear mapping. The effect of on coordinates is to map vectorsinto vectorsThis can be obtained by definingand performing the matrix multiplicationTherefore, is the matrix of the linear map with respect two the two bases and .

We still have to ascertain whether all linear maps have an associated matrix. It turns out that a map is linear if and only if it has one.

Proposition Let and be two linear spaces. Let be a basis for and a basis for . For any and any , denote by and their and coordinate vectors with respect to the two bases of and respectively. Let be a map. Then, is a linear map if and only if there exists an matrix such that, for any ,

Proof

Suppose such a matrix exists. Then, for any two vectors and any two scalars and , we have that where: in step we have used the properties of addition and scalar multiplication of coordinate vectors; in step we have applied the distributive property of matrix multiplication. We have just proved that if exists, then the mapping is linear. We now need to prove the converse statement (the "only if" part). Let be linear. Any element of the basis is transformed by into a vector that can be written as a linear combination of the basis as follows:where the scalars are the coefficients of the linear combination. Note that the coefficients are unique by the uniqueness of representations in terms of a basis. Denote by the matrix that is formed by all the coefficients , in such a way that the rows and columns of (indexed by and respectively) correspond to the different elements of the basis of and respectively. Now, take any and its associated coordinate vectorwhich means that can be written as a linear combination of the basis of as follows:Since is a linear map, we have thatThus, the coordinate vector of isWe have just proved that if is a linear map, then there exists a matrix of the map . This demonstrates the "only if" part of the proposition and concludes the proof.

We highly recommend reading the previous proof because it is a constructive proof that shows how to actually build the matrix by using the bases and . After going through the proof, we can see that the matrix is

In other words, the -th column of is the coordinate vector of the transformation of the -th vector of the basis .

The latter is a fact worth memorizing.

An important uniqueness result follows.

Proposition The matrix of a linear map with respect to two given bases is unique.

Proof

We have already demonstrated uniqueness in the equivalence proof above when we noted that the entries of the matrix are unique by the uniqueness of representations in terms of a basis.

Below you can find some exercises with explained solutions.

Let be the space of second-order polynomialswith basis . Let be the space of fourth-order polynomialswith basis . Define a linear mapping that transforms any polynomial into another polynomial equal to

Find the matrix .

Solution

The coordinate vector of the polynomial isThe polynomial is transformed as follows:Thus, the coordinate vector of the transformation iswhere is the matrix of the linear map with respect to the two bases.

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