The composition of two or more linear maps (also called linear functions or linear transformations) enjoys the same linearity property enjoyed by the two maps being composed. Moreover, the matrix of the composite transformation is equal to the product of the matrices of the two original maps.

Remember that a transformation (where and are vector spaces) is said to be a linear map if and only iffor any two vectors and any two scalars and .

We also need to remember that the composition of two functions and is a new function defined byfor any .

Example Suppose , and . Then,

The first important property of function composition is that it preserves linearity.

Proposition Let , and be three linear spaces. Let and be two functions. If and are linear maps, then also the composite transformation is a linear map.

Proof

Choose any two vectors and any two scalars and . Then,where: in step we have used the fact that is linear; in step we have used the linearity of . Thus, is linear.

Example Let , and be respectively spaces of , and column vectors having real entries. Define the map aswhere is a matrix, so that, for each , the product is a vector belonging to . Also define a map aswhere is a matrix, so that, for each , the product is a vector belonging to . In a previous lecture, we have proved that matrix multiplication defines linear maps on spaces of column vectors. As a consequence, and are linear maps. Thus, according to the previous proposition, the composite function is linear. That linearity holds can also be seen by directly computing the compositionwhere we can see that the matrix defines a linear transformation .

Remember that, given two linear spaces and , respectively endowed with two bases and , every linear map is associated to a matrix such that, for any ,where is the coordinate vector of with respect to the basis and is the coordinate vector of with respect to the basis . The matrix is called matrix of the linear map with respect to the bases and .

The next proposition shows that the composition of two linear maps is equivalent to multiplying their two matrices.

Proposition Let , and be three linear spaces endowed with bases , and respectively. Let and be two linear maps. Denote by the matrix of with respect to and . Denote by the matrix of with respect to and . Then, the composite function is the unique linear map such that

Proof

Take any . Then, maps into a vector whose coordinates are given bywhere the matrix is guaranteed to exist and is unique (see the lecture on the matrix of a linear map). Now, take and map it through into a vector having coordinateswhere the matrix is guaranteed to exist and is unique. By substituting (1) into (2), we obtainSince this is true for any , we have that the unique matrix productis the matrix of the linear map .

Below you can find some exercises with explained solutions.

Let , and be linear spaces respectively spanned by the bases

Let be a linear map such thatand be a linear map such that

Find the matrices , and .

Solution

The coordinate vectors of the transformed elements of the basis with respect to areandandThese coordinate vectors are the columns of the matrix of the transformation: The coordinate vectors of the transformed elements of the basis with respect to areandThus, we haveand

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