From A First Course in Linear Algebra
Version 2.23
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/
In this section we specialize and consider matrices with equal numbers of rows
and columns, which when considered as coefficient matrices lead to systems with
equal numbers of equations and variables. We will see in the second half of the
course (Chapter D, Chapter E Chapter LT, Chapter R) that these matrices are
especially important.
Our theorems will now establish connections between systems of equations (homogeneous or otherwise), augmented matrices representing those systems, coefficient matrices, constant vectors, the reduced rowechelon form of matrices (augmented and coefficient) and solution sets. Be very careful in your reading, writing and speaking about systems of equations, matrices and sets of vectors. A system of equations is not a matrix, a matrix is not a solution set, and a solution set is not a system of equations. Now would be a great time to review the discussion about speaking and writing mathematics in Technique L.
Definition SQM
Square Matrix
A matrix with m rows
and n columns is square
if m = n. In this case, we say
the matrix has size n.
To emphasize the situation when a matrix is not square, we will call it rectangular.
△
We can now present one of the central definitions of linear algebra.
Definition NM
Nonsingular Matrix
Suppose A is
a square matrix. Suppose further that the solution set to the homogeneous linear system
of equations ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right )
is \left \{0\right \},
i.e. the system has only the trivial solution. Then we say that
A is a nonsingular
matrix. Otherwise we say A
is a singular matrix. △
We can investigate whether any square matrix is nonsingular or not, no matter if the matrix is derived somehow from a system of equations or if it is simply a matrix. The definition says that to perform this investigation we must construct a very specific system of equations (homogeneous, with the matrix as the coefficient matrix) and look at its solution set. We will have theorems in this section that connect nonsingular matrices with systems of equations, creating more opportunities for confusion. Convince yourself now of two observations, (1) we can decide nonsingularity for any square matrix, and (2) the determination of nonsingularity involves the solution set for a certain homogeneous system of equations.
Notice that it makes no sense to call a system of equations nonsingular (the term does not apply to a system of equations), nor does it make any sense to call a 5 × 7 matrix singular (the matrix is not square).
Example S
A singular matrix, Archetype A
Example HISAA shows that the coefficient matrix derived from Archetype A, specifically
the 3 × 3
matrix,
A = \left [\array{
1&−1&2\cr
2& 1 &1
\cr
1& 1 &0 } \right ]

is a singular matrix since there are nontrivial solutions to the homogeneous system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ). ⊠
Example NM
A nonsingular matrix, Archetype B
Example HUSAB shows that the coefficient matrix derived from Archetype B, specifically
the 3 × 3
matrix,
B = \left [\array{
−7&−6&−12\cr
5 & 5 & 7
\cr
1 & 0 & 4 } \right ]

is a nonsingular matrix since the homogeneous system, ℒS\kern 1.95872pt \left (B,\kern 1.95872pt 0\right ), has only the trivial solution. ⊠
Notice that we will not discuss Example HISAD as being a singular or nonsingular coefficient matrix since the matrix is not square.
The next theorem combines with our main computational technique (rowreducing a matrix) to make it easy to recognize a nonsingular matrix. But first a definition.
Definition IM
Identity Matrix
The m × m identity
matrix, {I}_{m},
is defined by
(This definition contains Notation IM.) △
Example IM
An identity matrix
The 4 × 4
identity matrix is
{
I}_{4} = \left [\array{
1&0&0&0\cr
0&1 &0 &0
\cr
0&0&1&0\cr
0&0 &0 &1 } \right ].

Notice that an identity matrix is square, and in reduced rowechelon form. So in particular, if we were to arrive at the identity matrix while bringing a matrix to reduced rowechelon form, then it would have all of the diagonal entries circled as leading 1’s.
Theorem NMRRI
Nonsingular Matrices Row Reduce to the Identity matrix
Suppose that A is a
square matrix and B
is a rowequivalent matrix in reduced rowechelon form. Then
A is nonsingular if
and only if B is the
identity matrix. □
Proof ( ⇐) Suppose B is the identity matrix. When the augmented matrix \left [\left .A\kern 1.95872pt \right \vert \kern 1.95872pt 0\right ] is rowreduced, the result is \left [\left .B\kern 1.95872pt \right \vert \kern 1.95872pt 0\right ] = \left [\left .{I}_{n}\kern 1.95872pt \right \vert \kern 1.95872pt 0\right ]. The number of nonzero rows is equal to the number of variables in the linear system of equations ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ), so n = r and Theorem FVCS gives n − r = 0 free variables. Thus, the homogeneous system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ) has just one solution, which must be the trivial solution. This is exactly the definition of a nonsingular matrix.
( ⇒) If A is nonsingular, then the homogeneous system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ) has a unique solution, and has no free variables in the description of the solution set. The homogeneous system is consistent (Theorem HSC) so Theorem FVCS applies and tells us there are n − r free variables. Thus, n − r = 0, and so n = r. So B has n pivot columns among its total of n columns. This is enough to force B to be the n × n identity matrix {I}_{n} (see Exercise NM.T12). ■
Notice that since this theorem is an equivalence it will always allow us to determine if a matrix is either nonsingular or singular. Here are two examples of this, continuing our study of Archetype A and Archetype B.
Example SRR
Singular matrix, rowreduced
The coefficient matrix for Archetype A is
A = \left [\array{
1&−1&2\cr
2& 1 &1
\cr
1& 1 &0 } \right ]

which when rowreduced becomes the rowequivalent matrix
B = \left [\array{
\text{1}&0& 1\cr
0&\text{1 } &−1
\cr
0&0& 0 } \right ].

Since this matrix is not the 3 × 3 identity matrix, Theorem NMRRI tells us that A is a singular matrix. ⊠
Example NSR
Nonsingular matrix, rowreduced
The coefficient matrix for Archetype B is
A = \left [\array{
−7&−6&−12\cr
5 & 5 & 7
\cr
1 & 0 & 4 } \right ]

which when rowreduced becomes the rowequivalent matrix
B = \left [\array{
\text{1}&0&0\cr
0&\text{1 } &0
\cr
0&0&\text{1}} \right ].

Since this matrix is the 3 × 3 identity matrix, Theorem NMRRI tells us that A is a nonsingular matrix. ⊠
Nonsingular matrices and their null spaces are intimately related, as the next two examples illustrate.
Example NSS
Null space of a singular matrix
Given the coefficient matrix from Archetype A,
A = \left [\array{
1&−1&2\cr
2& 1 &1
\cr
1& 1 &0 } \right ]

the null space is the set of solutions to the homogeneous system of equations ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ) has a solution set and null space constructed in Example HISAA as
N\kern 1.95872pt \left (A\right ) = \left \{\left .\left [\array{
−{x}_{3}
\cr
{x}_{3}
\cr
{x}_{3} } \right ]\right \vert {x}_{3} ∈ {ℂ}^{}\right \}

Example NSNM
Null space of a nonsingular matrix
Given the coefficient matrix from Archetype B,
A = \left [\array{
−7&−6&−12\cr
5 & 5 & 7
\cr
1 & 0 & 4 } \right ]

the homogeneous system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ) has a solution set constructed in Example HUSAB that contains only the trivial solution, so the null space has only a single element,
N\kern 1.95872pt \left (A\right ) = \left \{\left [\array{
0\cr
0
\cr
0 } \right ]\right \}

These two examples illustrate the next theorem, which is another equivalence.
Theorem NMTNS
Nonsingular Matrices have Trivial Null Spaces
Suppose that A is a
square matrix. Then A
is nonsingular if and only if the null space of
A,
N\kern 1.95872pt \left (A\right ), contains only the
zero vector, i.e. N\kern 1.95872pt \left (A\right ) = \left \{0\right \}.
□
Proof The null space of a square matrix, A, is equal to the set of solutions to the homogeneous system, ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ). A matrix is nonsingular if and only if the set of solutions to the homogeneous system, ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ), has only a trivial solution. These two observations may be chained together to construct the two proofs necessary for each half of this theorem. ■
The next theorem pulls a lot of big ideas together. Theorem NMUS tells us that we can learn much about solutions to a system of linear equations with a square coefficient matrix by just examining a similar homogeneous system.
Theorem NMUS
Nonsingular Matrices and Unique Solutions
Suppose that A is
a square matrix. A
is a nonsingular matrix if and only if the system
ℒS\kern 1.95872pt \left (A,\kern 1.95872pt b\right )
has a unique solution for every choice of the constant vector
b.
□
Proof ( ⇐) The hypothesis for this half of the proof is that the system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt b\right ) has a unique solution for every choice of the constant vector b. We will make a very specific choice for b: b = 0. Then we know that the system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ) has a unique solution. But this is precisely the definition of what it means for A to be nonsingular (Definition NM). That almost seems too easy! Notice that we have not used the full power of our hypothesis, but there is nothing that says we must use a hypothesis to its fullest.
( ⇒) We assume that A is nonsingular of size n × n, so we know there is a sequence of row operations that will convert A into the identity matrix {I}_{n} (Theorem NMRRI). Form the augmented matrix {A}^{′} = \left [\left .A\kern 1.95872pt \right \vert \kern 1.95872pt b\right ] and apply this same sequence of row operations to {A}^{′}. The result will be the matrix {B}^{′} = \left [\left .{I}_{ n}\kern 1.95872pt \right \vert \kern 1.95872pt c\right ], which is in reduced rowechelon form with r = n. Then the augmented matrix {B}^{′} represents the (extremely simple) system of equations {x}_{i} ={ \left [c\right ]}_{i}, 1 ≤ i ≤ n. The vector c is clearly a solution, so the system is consistent (Definition CS). With a consistent system, we use Theorem FVCS to count free variables. We find that there are n − r = n − n = 0 free variables, and so we therefore know that the solution is unique. (This half of the proof was suggested by Asa Scherer.) ■
This theorem helps to explain part of our interest in nonsingular matrices. If a matrix is nonsingular, then no matter what vector of constants we pair it with, using the matrix as the coefficient matrix will always yield a linear system of equations with a solution, and the solution is unique. To determine if a matrix has this property (nonsingularity) it is enough to just solve one linear system, the homogeneous system with the matrix as coefficient matrix and the zero vector as the vector of constants (or any other vector of constants, see Exercise MM.T10).
Formulating the negation of the second part of this theorem is a good exercise. A singular matrix has the property that for some value of the vector b, the system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt b\right ) does not have a unique solution (which means that it has no solution or infinitely many solutions). We will be able to say more about this case later (see the discussion following Theorem PSPHS). Square matrices that are nonsingular have a long list of interesting properties, which we will start to catalog in the following, recurring, theorem. Of course, singular matrices will then have all of the opposite properties. The following theorem is a list of equivalences. We want to understand just what is involved with understanding and proving a theorem that says several conditions are equivalent. So have a look at Technique ME before studying the first in this series of theorems.
Theorem NME1
Nonsingular Matrix Equivalences, Round 1
Suppose that A
is a square matrix. The following are equivalent.
Proof That A is nonsingular is equivalent to each of the subsequent statements by, in turn, Theorem NMRRI, Theorem NMTNS and Theorem NMUS. So the statement of this theorem is just a convenient way to organize all these results. ■
Finally, you may have wondered why we refer to a matrix as nonsingular when it creates systems of equations with single solutions (Theorem NMUS)! I’ve wondered the same thing. We’ll have an opportunity to address this when we get to Theorem SMZD. Can you wait that long?
In Exercises C30–C33 determine if the matrix is nonsingular or singular. Give
reasons for your answer.
C30
\left [\array{
−3& 1 &2& 8\cr
2 & 0 &3 & 4
\cr
1 & 2 &7&−4\cr
5 &−1 &2 & 0
} \right ]

Contributed by Robert Beezer Solution [238]
\left [\array{
2 &3&1&4\cr
1 &1 &1 &0
\cr
−1&2&3&5\cr
1 &2 &1 &3 } \right ]

Contributed by Robert Beezer Solution [238]
\left [\array{
9& 3 &2& 4\cr
5&−6 &1 & 3
\cr
4& 1 &3&−5 } \right ]

Contributed by Robert Beezer Solution [238]
\left [\array{
−1& 2 & 0 &3\cr
1 &−3 &−2 &4
\cr
−2& 0 & 4 &3\cr
−3 & 1 &−2 &3
} \right ]

Contributed by Robert Beezer Solution [239]
C40 Each of the archetypes below is a system of equations with a square
coefficient matrix, or is itself a square matrix. Determine if these matrices are
nonsingular, or singular. Comment on the null space of each matrix.
Archetype A
Archetype B
Archetype F
Archetype K
Archetype L
Contributed by Robert Beezer
C50 Find the null space of the matrix E below.
Contributed by Robert Beezer Solution [239]
M30 Let A be the coefficient matrix of the system of equations below. Is A nonsingular or singular? Explain what you could infer about the solution set for the system based only on what you have learned about A being singular or nonsingular.
Contributed by Robert Beezer Solution [240]
For Exercises M51–M52 say as much as possible about each system’s
solution set. Be sure to make it clear which theorems you are using to reach your
conclusions.
M51 6 equations in 6 variables, singular coefficient matrix.
Contributed by Robert Beezer Solution [241]
M52 A system with a nonsingular coefficient matrix, not homogeneous.
Contributed by Robert Beezer Solution [241]
T10 Suppose that A is
a singular matrix, and B
is a matrix in reduced rowechelon form that is rowequivalent to
A. Prove that
the last row of B
is a zero row.
Contributed by Robert Beezer Solution [241]
T12 Suppose that A
is a square matrix. Using the definition of reduced rowechelon form
(Definition RREF) carefully, give a proof of the following equivalence: Every column of
A is a pivot column
if and only if A
is the identity matrix (Definition IM).
Contributed by Robert Beezer
T30 Suppose that A is a
nonsingular matrix and A is
rowequivalent to the matrix B.
Prove that B
is nonsingular.
Contributed by Robert Beezer Solution [242]
T90 Provide an alternative for the second half of the proof of
Theorem NMUS, without appealing to properties of the reduced
rowechelon form of the coefficient matrix. In other words, prove that if
A is nonsingular,
then ℒS\kern 1.95872pt \left (A,\kern 1.95872pt b\right )
has a unique solution for every choice of the constant vector
b.
Construct this proof without using Theorem REMEF or Theorem RREFU.
Contributed by Robert Beezer Solution [242]
C30 Contributed by Robert Beezer Statement [233]
The matrix rowreduces to
\left [\array{
\text{1}&0&0&0\cr
0&\text{1 } &0 &0
\cr
0&0&\text{1}&0\cr
0&0 &0 &\text{1} } \right ]

which is the 4 × 4 identity matrix. By Theorem NMRRI the original matrix must be nonsingular.
C31 Contributed by Robert Beezer Statement [233]
Rowreducing the matrix yields,
\left [\array{
\text{1}&0&0&−2\cr
0&\text{1 } &0 & 3
\cr
0&0&\text{1}&−1\cr
0&0 &0 & 0 } \right ]

Since this is not the 4 × 4 identity matrix, Theorem NMRRI tells us the matrix is singular.
C32 Contributed by Robert Beezer Statement [233]
The matrix is not square, so neither term is applicable. See Definition NM, which
is stated for just square matrices.
C33 Contributed by Robert Beezer Statement [234]
Theorem NMRRI tells us we can answer this question by simply rowreducing the
matrix. Doing this we obtain,
\left [\array{
\text{1}&0&0&0\cr
0&\text{1 } &0 &0
\cr
0&0&\text{1}&0\cr
0&0 &0 &\text{1} } \right ]

Since the reduced rowechelon form of the matrix is the 4 × 4 identity matrix {I}_{4}, we know that B is nonsingular.
C50 Contributed by Robert Beezer Statement [235]
We form the augmented matrix of the homogeneous system
ℒS\kern 1.95872pt \left (E,\kern 1.95872pt 0\right ) and
rowreduce the matrix,
We knew ahead of time that this system would be consistent (Theorem HSC), but we can now see there are n − r = 4 − 2 = 2 free variables, namely {x}_{3} and {x}_{4} since F = \left \{3, 4, 5\right \} (Theorem FVCS). Based on this analysis, we can rearrange the equations associated with each nonzero row of the reduced rowechelon form into an expression for the lone dependent variable as a function of the free variables. We arrive at the solution set to this homogeneous system, which is the null space of the matrix by Definition NSM,
M30 Contributed by Robert Beezer Statement [235]
We rowreduce the coefficient matrix of the system of equations,
Since the rowreduced version of the coefficient matrix is the 4 × 4 identity matrix, {I}_{4} (Definition IM byTheorem NMRRI, we know the coefficient matrix is nonsingular. According to Theorem NMUS we know that the system is guaranteed to have a unique solution, based only on the extra information that the coefficient matrix is nonsingular.
M51 Contributed by Robert Beezer Statement [236]
Theorem NMRRI tells us that the coefficient matrix will not rowreduce to
the identity matrix. So if we were to rowreduce the augmented matrix
of this system of equations, we would not get a unique solution. So by
Theorem PSSLS the remaining possibilities are no solutions, or infinitely
many.
M52 Contributed by Robert Beezer Statement [236]
Any system with a nonsingular coefficient matrix will have a unique solution by
Theorem NMUS. If the system is not homogeneous, the solution cannot be the
zero vector (Exercise HSE.T10).
T10 Contributed by Robert Beezer Statement [236]
Let n denote the size
of the square matrix A.
By Theorem NMRRI the hypothesis that
A is singular implies
that B is not the
identity matrix {I}_{n}.
If B has
n pivot columns, then
it would have to be {I}_{n},
so B must have
fewer than n
pivot columns. But the number of nonzero rows in
B
(r)
is equal to the number of pivot columns as well. So the
n rows of
B have fewer
than n nonzero
rows, and B
must contain at least one zero row. By Definition RREF, this row must be at the
bottom of B.
T30 Contributed by Robert Beezer Statement [236]
Since A
and B
are rowequivalent matrices, consideration of the three row
operations (Definition RO) will show that the augmented matrices,
\left [\left .A\kern 1.95872pt \right \vert \kern 1.95872pt 0\right ] and
\left [\left .B\kern 1.95872pt \right \vert \kern 1.95872pt 0\right ], are
also rowequivalent matrices. This says that the two homogeneous systems,
ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ) and
ℒS\kern 1.95872pt \left (B,\kern 1.95872pt 0\right ) are equivalent
systems. ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right )
has only the zero vector as a solution (Definition NM), thus
ℒS\kern 1.95872pt \left (B,\kern 1.95872pt 0\right ) has
only the zero vector as a solution. Finally, by Definition NM, we see that
B is
nonsingular.
Form a similar theorem replacing “nonsingular” by “singular” in both the hypothesis and the conclusion. Prove this new theorem with an approach just like the one above, and/or employ the result about nonsingular matrices in a proof by contradiction.
T90 Contributed by Robert Beezer Statement [237]
We assume A
is nonsingular, and try to solve the system
ℒS\kern 1.95872pt \left (A,\kern 1.95872pt b\right ) without making any
assumptions about b.
To do this we will begin by constructing a new homogeneous linear
system of equations that looks very much like the original. Suppose
A has
size n
(why must it be square?) and write the original system as,
Form the new, homogeneous system in n equations with n + 1 variables, by adding a new variable y, whose coefficients are the negatives of the constant terms,
Since this is a homogeneous system with more variables than equations (m = n + 1 > n), Theorem HMVEI says that the system has infinitely many solutions. We will choose one of these solutions, any one of these solutions, so long as it is not the trivial solution. Write this solution as
We know that at least one value of the {c}_{i} is nonzero, but we will now show that in particular {c}_{n+1}\mathrel{≠}0. We do this using a proof by contradiction (Technique CD). So suppose the {c}_{i} form a solution as described, and in addition that {c}_{n+1} = 0. Then we can write the ith equation of system (∗∗) as,
}
Since this is true for each i, we have that {x}_{1} = {c}_{1},\kern 1.95872pt {x}_{2} = {c}_{2},\kern 1.95872pt {x}_{3} = {c}_{3},\mathop{\mathop{…}},\kern 1.95872pt {x}_{n} = {c}_{n} is a solution to the homogeneous system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ) formed with a nonsingular coefficient matrix. This means that the only possible solution is the trivial solution, so {c}_{1} = 0,\kern 1.95872pt {c}_{2} = 0,\kern 1.95872pt {c}_{3} = 0,\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {c}_{n} = 0. So, assuming simply that {c}_{n+1} = 0, we conclude that all of the {c}_{i} are zero. But this contradicts our choice of the {c}_{i} as not being the trivial solution to the system (∗∗). So {c}_{n+1}\mathrel{≠}0.
We now propose and verify a solution to the original system (∗). Set
Notice how it was necessary that we know that {c}_{n+1}\mathrel{≠}0 for this step to succeed. Now, evaluate the ith equation of system (∗) with this proposed solution, and recognize in the third line that {c}_{1} through {c}_{n+1} appear as if they were substituted into the lefthand side of the ith equation of system (∗∗),
Since this equation is true for every i, we have found a solution to system (∗). To finish, we still need to establish that this solution is unique.
With one solution in hand, we will entertain the possibility of a second solution. So assume system (∗) has two solutions,
Then,
This is the ith equation of the homogeneous system ℒS\kern 1.95872pt \left (A,\kern 1.95872pt 0\right ) evaluated with {x}_{j} = {d}_{j} − {e}_{j}, 1 ≤ j ≤ n. Since A is nonsingular, we must conclude that this solution is the trivial solution, and so 0 = {d}_{j} − {e}_{j}, 1 ≤ j ≤ n. That is, {d}_{j} = {e}_{j} for all j and the two solutions are identical, meaning any solution to (∗) is unique.
Notice that the proposed solution ({x}_{i} = {{c}_{i}\over { c}_{n+1}}) appeared in this proof with no motivation whatsoever. This is just fine in a proof. A proof should convince you that a theorem is true. It is your job to read the proof and be convinced of every assertion. Questions like “Where did that come from?” or “How would I think of that?” have no bearing on the validity of the proof.
At the conclusion of each chapter you will find a section like this, reviewing selected definitions and theorems. There are many reasons for why a definition or theorem might be placed here. It might represent a key concept, it might be used frequently for computations, provide the critical step in many proofs, or it may deserve special comment.
These lists are not meant to be exhaustive, but should still be useful as part of reviewing each chapter. We will mention a few of these that you might eventually recognize on sight as being worth memorization. By that we mean that you can associate the acronym with a rough statement of the theorem — not that the exact details of the theorem need to be memorized. And it is certainly not our intent that everything on these lists is important enough to memorize.
Theorem RCLS
We will repeatedly appeal to this theorem to determine if a system of linear
equations, does, or doesn’t, have a solution. This one we will see often enough
that it is worth memorizing.
Theorem HMVEI
This theorem is the theoretical basis of several of our most important theorems.
So keep an eye out for it, and its descendants, as you study other proofs. For
example, Theorem HMVEI is critical to the proof of Theorem SSLD,
Theorem SSLD is critical to the proof of Theorem G, Theorem G is critical to
the proofs of the pair of similar theorems, Theorem ILTD and Theorem SLTD,
while finally Theorem ILTD and Theorem SLTD are critical to the proof of an
important result, Theorem IVSED. This chain of implications might not
make much sense on a first reading, but come back later to see how some
very important theorems build on the seemingly simple result that is
Theorem HMVEI. Using the “find” feature in whatever software you use to
read the electronic version of the text can be a fun way to explore these
relationships.
Theorem NMRRI
This theorem gives us one of simplest ways, computationally, to recognize if a
matrix is nonsingular, or singular. We will see this one often, in computational
exercises especially.
Theorem NMUS
Nonsingular matrices will be an important topic going forward (witness the
NMEx series of theorems). This is our first result along these lines, a useful
theorem for other proofs, and also illustrates a more general concept from
Chapter LT.