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Composition of linear maps

by , PhD

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.

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Preliminaries

Remember that a transformation $f:S ightarrow T$ (where $S$ and $T$ are vector spaces) is said to be a linear map if and only if[eq1]for any two vectors $s_{1},s_{2}in S$ and any two scalars $lpha _{1}$ and $lpha _{2}$.

We also need to remember that the composition of two functions $f:S ightarrow T$ and $g:T ightarrow U$ is a new function [eq2] defined by[eq3]for any $sin S$.

Example Suppose $S=T=U=U{211d} $, [eq4] and [eq5]. Then,[eq6]

Composition preserves linearity

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

Proposition Let $S$, $T$ and $U$ be three linear spaces. Let $f:S ightarrow T$ and $g:T ightarrow U$ be two functions. If $f$ and $g$ are linear maps, then also the composite transformation [eq7] is a linear map.

Proof

Choose any two vectors $s_{1},s_{2}in S$ and any two scalars $lpha _{1}$ and $lpha _{2}$. Then,[eq8]where: in step $ rame{A}$ we have used the fact that $f$ is linear; in step $ rame{B}$ we have used the linearity of $g$. Thus, $gcirc f$ is linear.

Example Let $S$, $T$ and $U$ be respectively spaces of $3 imes 1$, $2 imes 1$ and $3 imes 1$ column vectors having real entries. Define the map $f:S ightarrow T$ as[eq9]where F is a $2 imes 3$ matrix, so that, for each $sin S$, the product $Fs$ is a $2 imes 1$ vector belonging to $T$. Also define a map $g:T ightarrow U$ as[eq10]where $G$ is a $3 imes 2$ matrix, so that, for each $tin T$, the product $Gt$ is a $3 imes 1$ vector belonging to $U$. In a previous lecture, we have proved that matrix multiplication defines linear maps on spaces of column vectors. As a consequence, $f$ and $g$ are linear maps. Thus, according to the previous proposition, the composite function $gcirc f $ is linear. That linearity holds can also be seen by directly computing the composition[eq11]where we can see that the $3 imes 3$ matrix $GF$ defines a linear transformation [eq12].

The matrix of the composition

Remember that, given two linear spaces $S$ and $T$, respectively endowed with two bases $B$ and $C$, every linear map $f:S ightarrow T$ is associated to a matrix [eq13] such that, for any $sin S$,[eq14]where [eq15] is the coordinate vector of $s$ with respect to the basis $B$ and [eq16] is the coordinate vector of $fleft( s ight) $ with respect to the basis $C$. The matrix [eq17] is called matrix of the linear map with respect to the bases $B$ and $C$.

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

Proposition Let $S$, $T$ and $U$ be three linear spaces endowed with bases $B$, $C$ and $D$ respectively. Let $f:S ightarrow T$ and $g:T ightarrow U$ be two linear maps. Denote by [eq18] the matrix of $f$ with respect to $B$ and $C$. Denote by [eq19] the matrix of $g$ with respect to $C$ and $D$. Then, the composite function $gcirc f$ is the unique linear map such that[eq20]

Proof

Take any $sin S$. Then, $f$ maps $s$ into a vector [eq21] whose coordinates are given by[eq22]where the matrix [eq23] is guaranteed to exist and is unique (see the lecture on the matrix of a linear map). Now, take $fleft( s ight) $ and map it through $g$ into a vector [eq24] having coordinates[eq25]where the matrix [eq26] is guaranteed to exist and is unique. By substituting (1) into (2), we obtain[eq27]Since this is true for any $sin S$, we have that the unique matrix product[eq28]is the matrix of the linear map [eq29].

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let $S$, $T$ and $U$ be linear spaces respectively spanned by the bases [eq30]

Let $f:S ightarrow T$ be a linear map such that[eq31]and $g:T ightarrow U$ be a linear map such that[eq32]

Find the matrices [eq23], [eq26] and [eq35].

Solution

The coordinate vectors of the transformed elements of the basis $B$ with respect to $C$ are[eq36]and[eq37]and[eq38]These coordinate vectors are the columns of the matrix of the transformation: [eq39]The coordinate vectors of the transformed elements of the basis $C$ with respect to $D$ are[eq40]and[eq41]Thus, we have[eq42]and[eq43]

How to cite

Please cite as:

Taboga, Marco (2021). "Composition of linear maps", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/composition-of-linear-maps.

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