Linear Algebra

228 Chapter 3. Maps Between Spaces
We have observed the following result, which we shall use in the next subsection.
3.22 Corollary For any matrix H there are elementary reduction matrices
R1, ... , Rr such that Rr · Rr−1 ···R1 · H is in reduced echelon form.
Until now we have taken the point of view that our primary objects of study
are vector spaces and the maps between them, and have adopted matrices only
for computational convenience. This subsection show that this point of view
isn’t the whole story. Matrix theory is a fascinating and fruitful area.
In the rest of this book we shall continue to focus on maps as the primary
objects, but we will be pragmatic—if the matrix point of view gives some clearer
idea then we shall use it.
Exercises
� 3.23 Predict the result of each multiplication by an elementary reduction matrix,
and then � check ��by multiplying � it�out. �� � � �� �
3 0 1 2
4 0 1 2
1 0 1 2
(a)
(b)
(c)
0 0 3 4
0 2 3 4
−2 1 3 4
� �� � � �� �
1 2 1 −1
1 2 0 1
(d)
(e)
3 4 0 1
3 4 1 0
� 3.24 The need to take linear combinations of rows and columns in tables of numbers
arises often in practice. For instance, this is a map of part of Vermont and New
York.
In part because of Lake Champlain,
there are no roads connecting some
pairs of towns. For instance, there
isnowaytogofromWinooskito
Grand Isle without going through
Colchester. (Of course, many other
roads and towns have been left off
to simplify the graph. From top to
bottom of this map is about forty
miles.)
Grand Isle
Swanton
Colchester
Winooski
Burlington
(a) The incidence matrix of a map is the square matrix whose i, j entry is the
number of roads from city i to city j. Produce the incidence matrix of this map
(take the cities in alphabetical order).
(b) Amatrixissymmetric if it equals its transpose. Show that an incidence
matrix is symmetric. (These are all two-way streets. Vermont doesn’t have
many one-way streets.)
(c) What is the significance of the square of the incidence matrix? The cube?