Composing maps

For column vectors, applying B first and A second is represented by AB, because A(Bx) equals (AB)x.

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On this page 7 sections

Overview
All vectors here are columns
Let A double the first coordinate and B swap coordinates
Check your understanding
Key takeaway
Sources & further reading
Concept connections

01All vectors here are columns#

All vectors here are columns. If a vector passes through $B$ first and $A$ second, its output is $A(Bx)$ .

The composition matrix is

AB

The first map sits nearest the input vector, on the right.

Use $A=\begin{pmatrix}2&0\\0&1\end{pmatrix}$ and $B=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ .

First B, then A:

AB=\begin{pmatrix}0&2\\1&0\end{pmatrix}

Entry $(1,2)$ is $2(1)+0(0)=2$ .

For compatible matrices and a column vector $x$ :

Associativity groups the stages:

A(Bx)=(AB)x

If $B$ is $p\times n$ and $A$ is $m\times p$ , then $AB$ is $m\times n$ .

Check your reasoning

$A=\begin{pmatrix}3&0\\0&1\end{pmatrix}$ , $B=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ . First $A$ , then $B$ . Find entry $(2,1)$ of the composition matrix.

Show answer and explanation

The product is $BA$ . Its entry $(2,1)$ is $1(3)+0(0)=3$ .

02Let A double the first coordinate and B swap coordinates#

Let $A$ double the first coordinate and $B$ swap coordinates. Starting at $x=(1,3)$ , reversing the maps changes the result.

A doubles the first coordinate; B swaps coordinates. From x=(1,3), B then A reaches (3,1) then (6,1). A then B reaches (2,3) then (3,2). Lines connect successive states, not additional vector sums. — Figure 1The same input gives different outputs in the two orders.

Check your reasoning

Column vectors pass through $S$ ( $3\times2$ ), then $P$ ( $1\times3$ ). Choose the product.

A $SP$
B $PS$
C $P+S$

Show answer and explanation

$PS$

Use $PS$ : $S$ acts first.

03Check your understanding#

Check your reasoning

$A$ : $2\times3$ , $B$ : $3\times2$ . A column vector goes through $A$ , then $B$ . Repair the claim $AB$ .

A $AB$ : $2\times2$
B $BA$ : $2\times2$
C $BA$ : $3\times3$

Show answer and explanation

$BA$ : $3\times3$

Use $BA$ : 3 inputs and 3 outputs.

Key takeaway

With column vectors, the rightmost map acts first.

Write the matrix product in the order the maps act on a column vector.

Sources & further reading

[1]
Boyd and Vandenberghe, Introduction to Applied Linear Algebra (2018), 10.2 Composition of linear functions, p.183 ↗Boyd and Vandenberghe · Book

01All vectors here are columns#

02Let A double the first coordinate and B swap coordinates#

03Check your understanding#

With column vectors, the rightmost map acts first.

Sources & further reading

Reference this concept