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Section 4.1 Matrices and Multiplication (MX1)

Subsection 4.1.1 Warm Up

Activity 4.1.1.

Suppose that T:VW is a linear transformation.
(a)
What is the definition of kerT? How does it relate to the codomain of T?
(b)
What is definition of ImT? How does it relate to the codomain of T?

Subsection 4.1.2 Class Activities

Observation 4.1.2.

If T:RnRm and S:RmRk are linear maps, then the composition map ST computed as (ST)(v)=S(T(v)) is a linear map from RnRk.
Figure 40. The composition of two linear maps.

Activity 4.1.3.

Let T:R3R2 be defined by the 2×3 standard matrix B and S:R2R4 be defined by the 4×2 standard matrix A:
B=[213534]A=[12013512].
(a)
What are the domain and codomain of the composition map ST?
  1. The domain is R3 and the codomain is R2
  2. The domain is R2 and the codomain is R4
  3. The domain is R3 and the codomain is R4
  4. The domain is R4 and the codomain is R3
(b)
What size will the standard matrix of ST be?
  1. 4 (rows)×3 (columns)
  2. 3 (rows)×4 (columns)
  3. 3 (rows)×2 (columns)
  4. 2 (rows)×4 (columns)
(c)
Compute
(ST)(e1)=S(T(e1))=S([25])=[????].
(f)
Use (ST)(e1),(ST)(e2),(ST)(e3) to write the standard matrix for ST.

Definition 4.1.4.

We define the product AB of a m×n matrix A and a n×k matrix B to be the m×k standard matrix of the composition map of the two corresponding linear functions.
For the previous activity, T was a map R3R2, and S was a map R2R4, so ST gave a map R3R4 with a 4×3 standard matrix:
AB=[12013512][213534]
=[(ST)(e1)(ST)(e2)(ST)(e3)]=[12555343112111255].

Activity 4.1.5.

Let S:R3R2 be given by the matrix A=[423011] and T:R2R3 be given by the matrix B=[231101].
(a)
Write the dimensions (rows × columns) for A, B, AB, and BA.
(b)
Find the standard matrix AB of ST.
(c)
Find the standard matrix BA of TS.

Activity 4.1.6.

Consider the following three matrices.
A=[103321]B=[22101111100032115721]C=[22013140]
(a)
Find the domain and codomain of each of the three linear maps corresponding to A, B, and C.
(b)
Only one of the matrix products AB,AC,BA,BC,CA,CB can actually be computed. Compute it.

Activity 4.1.7.

Let B=[340201033], and let A=[271032111].
(a)
Compute the product BA by hand.
(b)
Check your work using technology. Using Octave:
B = [3 -4 0 ; 2 0 -1 ; 0 -3 3]
A = [2 7 -1 ; 0 3 2  ; 1 1 -1]
B*A
    

Activity 4.1.8.

Of the following three matrices, only two may be multiplied.
A=[13231423]B=[161010]C=[111012241231]
Explain which two can be multiplied and why. Then show how to find their product.

Activity 4.1.9.

Let T([xy])=[x+2yy3x+5yx2y] In Fact 3.2.12 we adopted the notation
T([xy])=[x+2yy3x+5yx2y]=A[xy]=[12013512][xy].
Verify that [12013512][xy]=[x+2yy3x+5yx2y] in terms of matrix multiplication.

Subsection 4.1.3 Individual Practice

Activity 4.1.10.

Given two n×n matrices A and B, explain why the sentence "Multiply the matrices A and B together." is ambiguous. How could you re-write the sentence in order to eliminate the ambiguity?

Subsection 4.1.4 Videos

Figure 41. Video: Multiplying matrices

Exercises 4.1.5 Exercises

Subsection 4.1.6 Mathematical Writing Explorations

Exploration 4.1.11.

Construct 3 matrices, A,B, and C, such that
  • AB:R4R2
  • BC:R2R3
  • CA:R3R4
  • ABC:R2R2

Exploration 4.1.12.

Construct 3 examples of matrix multiplication, with all matrix dimensions at least 2.
  • Where A and B are not square, but AB is square.
  • Where AB=BA.
  • Where ABBA.

Exploration 4.1.13.

Use the included map in this problem.
A map with 5 dots. A is connected to B, B is connected to C, C is connected to D and E, and D and E are connected to each other
Figure 42. Adjacency map, showing roads between 5 cities
  • An adjacency matrix for this map is a matrix that has the number of roads from city i to city j in the (i,j) entry of the matrix. A road is a path of length exactly 1. All (i,i)entries are 0. Write the adjacency matrix for this map, with the cities in alphabetical order.
  • What does the square of this matrix tell you about the map? The cube? The n-th power?

Subsection 4.1.7 Sample Problem and Solution

Sample problem Example B.1.18.