Linear maps are transformations from one vector space to another that have the property of preserving vector addition and scalar multiplication.

Let us start with a definition.

Definition Let and be two linear spaces. Let be a transformation that associates one and only one element of to each element of . The transformation is said to be a linear map if and only iffor any two scalars and and any two vectors .

While "map" is probably the most commonly used term, we can interchangeably use the terms "mapping", "transformation" and "function".

Example Let be the space of all column vectors having real entries. Let be the space of column vectors having real entries. Suppose the map associates to each vector a vectorNow, take any two vectors and any two scalars and . By repeatedly applying the definitions of vector addition and scalar multiplication, we getThus, is a linear map.

We will later prove that every linear map can be represented by a matrix, but the converse is also true: pre-multiplication of vectors by a matrix defines a linear map.

Proposition Let be the linear space of all column vectors. Let be the linear space of all column vectors. Let be a matrix. Consider the transformation defined, for any , by where denotes the matrix product between and . Then is a linear map.

Proof

For any two vectors and any two scalars and , we have that where in step we have applied the distributive property of matrix multiplication.

The same result holds for post-multiplication.

Proposition Let be the linear space of all row vectors. Let be the linear space of all row vectors. Let be a matrix. Consider the transformation defined, for any , by where denotes the matrix product between and . Then is a linear map.

Proof

Analogous to the previous proof.

As it might be intuitive to understand, linear maps preserve the linearity also of combinations that involve more than two terms.

Proposition Let be scalars and let be elements of a linear space . If is a linear map, then

Proof

The result is obtained by applying the linearity property to one vector at a time:

A very interesting and useful property is that a linear map is completely determined by its values on a basis of (i.e., a set of linearly independent vectors such that any vector can be written as a linear combination of the basis).

Proposition Let and be linear spaces. Let be a basis of . Let . Then, there is a unique linear map such thatfor .

Proof

Any vector can be written as a linear combination of the basis:where the scalars are unique because representations in terms of a basis are unique. Then, the linearity of the map implies that for any . Thus, the value of the map is uniquely determined by the vectors (which were chosen in advance) and by the unique scalars .

In other words, if we know the values taken by the map in correspondence to the vectors of the basis, then we are able to derive also all the other values taken by the map.

Example Let be the space of all vectors. Let be the space of all vectors. Consider the linear map such thatThe two vectorsform a basis for (the canonical basis of ). Any vector can be written as a linear combination of the basis. In particular, if we denote by and the two entries of , then we have thatTherefore, the value of in correspondence of any vector can be derived as follows:

Below you can find some exercises with explained solutions.

Let be the space of all vectors. Define the function that maps each vector as follows:Determine whether is a linear map.

Solution

Take any two vectors and any two scalars and . We have thatThe map would be linear if the vector was equal to zero for any two scalars and . But the latter vector is different from zero for any choice of and such that . Therefore, the map is not linear.

Let be the space of all vectors. Define the function that maps each vector as follows:Determine whether is a linear map.

Solution

For any two vectors and any two scalars and , we have thatThus, the map is linear.

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