:: Additional Practise Problems for Discover Linear Algebra

All lists of problems are from suggested supplementary textbook:

3,000 Solved Problems in Linear Algebra
Lipschutz, Seymour
McGraw-Hill Education (1989)
ISBN-10: 0070380236
ISBN-13: 978-0070380233

Additional practise problems are provided in linked pages.

NOTE   See the initial Practise Problems page for practise problems related to chapters that are shared between the one-semester and two-semester versions.

Appendix A Complex Numbers

Problems §1.9 # 1.113-1.117, 1.120, 1.122, 1.125.

Chapter 22 Change of basis

Problems §13.1 # 13.3–5, 13.6–12, 13.14–17, 13.19–29, 13.34–39.
Comments Terminology: The supplementary text reverses the designations of "old" and "new" basis from the usage in Discover Linear Algebra: what they call "new" we would call "old," and vice versa. Their point of view is based on the linear formulas in the solution to problem 13.1, whereas our point of view is based on matrix formula (i) in their Theorem 13.2.
§13.1 #13.3: You don't need to use #13.2 as they do in their solution, you should set up an appropriate matrix and row reduce.
§13.1 #13.7: You don't need to use #13.6 as they do in their solution, you should set up an appropriate matrix and row reduce.
§13.1 #13.11,12: Maybe just do these for some example values of a,b, instead of their general situation.
§13.1 #13.14: You don't need to use #13.13 as they do in their solution, you should set up an appropriate matrix and row reduce.
§13.1 #13.19: You don't need to use #13.18 as they do in their solution, you should set up an appropriate matrix and row reduce.
§13.1 #13.24,25,30: Maybe just do these for some example values of a,b,c, instead of their general situation.
§13.1 #13.39: Maybe just do these for some example values of a,b, instead of their general situation.

Chapter 26 Similarity

Problems §13.3: # 13.62–65, 13.71–73.

Chapter 27 Application to systems of differential equations

Problems §21.5 All.

Chapter 28 Block-diagonal form

Problems §4.16 # 4.266–269, 4.4.273–275,
§5.9 # 5.148–151.
§16.1 # 16.15–16.19.
§17.1 # 17.1–6, 17.7–9, 17.12, 17.17.
§17.2 # 17.25 But answer this question instead: Do U and W form a complete set of independent subspaces?
§17.2 # 17.26 But answer this question instead: Do U and Z form a complete set of independent subspaces?
Comments §17: In all these questions, take \(V=\mathbb{R}^n\) (or possibly \(V=\mathbb{C}^n\)), and take "linear operator T" (or S or whatever letter) as "multiplication of vectors in \(\mathbb{R}^n\) by the matrix T" instead.
§17 # 17.9: Replace the word "kernel" by "null space." (Remember we are thinking of T as a matrix, not a "linear operator," whatever that is.)

Chapter 29 Scalar-triangular form

See the following pages.

Practise problems for scalar-triangular form
Answers to practise problems for scalar-triangular form

Chapter 30 Triangular block form

See the following pages.

Practise problems for triangular block form
Answers to practise problems for triangular block form

Chapter 31 Consequences of triangular block form

See the following pages.

Practise problems for consequences of triangular block form
Answers to practise problems for consequences of triangular block form

Chapter 32 Elementary nilpotent form

Problems §17.4 # 17.47, 17.49, 17.50–56, 17.62, 17.64–65, 17.71, 17.72, 17.74, 17.75.
Comments Terminology:
  • In any question that uses the word "operator," just ignore that word and concentrate on the matrix version of the question.
  • What we call the "degree of nilpotency," this book calls the "index of nilpotency."
  • What we call "elementary nilpotent form" they call a "basic nilpotent block," and they also use the upper triangular version whereas we use the lower triangular version.
  • Their solutions frequently rely on the Cayley-Hamilton Theorem, but in our treatment in the online textbook we proved a bunch of properties of nilpotent matrices before proving the Cayley-Hamilton Theorem, so logically we couldn't rely on the C-H Theorem to prove those nilpotent properties.
§17.4 # 17.64–65: In Lemma 17.13 preceding these questions, consider \(T\) to be an \(n \times n\) matrix, and \(v\) to be a vector in \(\mathbb{R}^n\).

The following pages contain extra practise problems and their answers.

Extra practise problems for elementary nilpotent form
Answers to extra practise problems for elementary nilpotent form

Chapter 33 Triangular-block nilpotent form

See the following pages.

Practise problems for triangular-block nilpotent form
Answers to practise problems for triangular-block nilpotent form

Chapter 34 Jordan normal form

Problems §17.5 # 17.76–85, 17.88–90, 17.92–93, 17.95–97, 17.99, 17.101–110, 17.112–113.
Comments Terminology:
  • In any question that uses the word "operator," just ignore that word and concentrate on the matrix version of the question.
  • What we call the "Jordan normal form," this book calls the "Jordan canonical form," and their Jordan blocks are the upper triangular version whereas we have opted to use the lower triangular version.
  • What we call "elementary nilpotent form" they call a "basic nilpotent block," and they also use the upper triangular version whereas we use the lower triangular version.
§17.5 # 17.79: Ignore the bit about "minimal polynomial."
§17.5 # 17.84: In our version of what a Jordan normal form matrix is, we wouldn't separate the two \(\lambda = 5\) blocks.
§17.5 # 17.101–104: Ignore the bit about "minimal polynomial," just determine all possible Jordan normal form matrices that have the given characteristic polynomial. (And then, since we are using less information to answer the question, you should come up with more possible answers than they do.)
§17.5 # 17.105–110: Ignore the bit about "minimal polynomial."

The following pages contain extra practise problems and their answers.

Extra practise problems for elementary nilpotent form
Answers to extra practise problems for elementary nilpotent form

Chapter 36 Abstract inner product spaces

Problems §14.1: Choose a selection of questions to hit different examples of inner products on different types of spaces, and involving different types of calculations (inner product, norm, angle). You may skip questions on dot product if you wish, since that is old material.
§14.2 # 14.66, 14.67, 14.69, 14.71.
§14.3 # 14.81, 14.82, 14.85, 14.86. A selection of problems from 14.87–91 to practise determining angles between vectors.
§14.8 # 14.199, 14.209–14.216.
§14.9 # 14.217–225, 14.226(a), 14.227–231.
Comments Only §14.9 deals with complex inner products; in the other sections, you should assume a real inner product.

Chapter 37 Orthogonality

Problems §14.4 All except # 14.104, 14.111.
§14.5 All except # 14.137, 14.148.
§14.7 # 14.177, 14.183, 14.185–187, 14.189–192, 14.194.
§14.9 # 14.234.
Comments Only §14.9 deals with complex inner products; in the other sections, you should assume a real inner product.

Chapter 38 Orthogonal projection and best approximation

Problems §14.7 # 14.178–181.
Comments §14.7: Here "Fourier coefficient" just means the projection coefficient.

The following pages contain extra practise problems and their answers.

Extra practise problems for best approximation
Answers to extra practise problems for best approximation

Chapter 39 Matrix adjoints

Problems §14.6 # 14.155–14.166, 14.168–172.
§20.1 # 20.3–6, 20.14–22.
§20.2 # 20.25, 20.40–42.
§20.3 # 20.47, 20.53–56.
Comments §20: In any of these questions that do not involve a specific matrix, you should assume that:
  • \(V\) means either \(\mathbb{R}^n\) or \(\mathbb{C}^n\) (real vs complex contexts).
  • "Linear operator \(T\)" means that \(T\) is an \(n \times n\) matrix.
  • Vectors are column vectors.
  • Expressions like \(T(v)\) mean the result of multiplication of square matrix \(T\) times column vector \(v\).

The following page contains extra practise problems.

Extra practise problems for matrix adjoints.

Chapter 40 Orthogonal/unitary diagonalization

Problems §14.6 # 14.173–176.
§20.2 # 20.26–28, 20.30–36.
§20.3 # 20.52.
§20.5 # 20.78–86.
Comments §14.6: The supplementary book uses the phrase "orthogonally equivalent" to mean what we might call "orthogonally similar."
§20: In any of these questions that do not involve a specific matrix, you should assume that:
  • \(V\) means either \(\mathbb{R}^n\) or \(\mathbb{C}^n\) (real vs complex contexts).
  • "Linear operator \(T\)" means that \(T\) is an \(n \times n\) matrix.
  • Vectors are column vectors.
  • Expressions like \(T(v)\) mean the result of multiplication of square matrix \(T\) times column vector \(v\).

The following page contains extra practise problems.

Extra practise problems for orthogonal/unitary diagonalization.

Chapter 41 Quadratic forms

Problems §19.5 # 19.59–63, 19.65–72.
§19.7 # 19.98, 19.99, 19.101, 19.102, 19.104, 19.105–110, 19.112.

Chapter 42 Matrix and linear transformations

Problems §9.1 # 9.1–3, 9.7, 9.9, 9.15, 9.18–20, 9.33–35, 19.59–63, 19.65–72.
§9.3 # 9.73–77, 9.89–91, 9.93, 9.94, 9.97, 9.98.
§10.1 All.
§10.2 # 10.29–31, 10.33–35, 10.37–39, 10.42–45.
§11.1 # 11.1–10, 11.15, 11.17, 11.19–21, 11.32.
§11.2 # 11.33–43, 11.45, 11.46.
§11.5 # 11.133.
§12.1 # 12.27, 12.33, 12.35, 12.36, 12.41–43.
§12.2 # 12.51–53.
§12.4 # 12.125–129.
§18.1 All.

Chapter 43 Kernel and image

Problems §9.1 # 9.4, 9.5, 9.11, 9.16, 9.30.
§9.3 # 9.87, 9.96.
§10.3 All except # 10.59, 10.71–73.
§10.4 All.
§11.5 # 11.134.

Chapter 44 Compositions, inverses, and isomorphisms

Problems §9.1 # 9.13.
§9.3 # 9.78, 9.82, 9,83, 9.84, 9.86, 9.92.
§9.4 Some selection of problems from # 9.99–126 to remind yourself how composition of functions works.
§9.5 Some selection of problems from # 9.134–163 to remind yourself how one-to-one (injective) and onto (surjective) work.
§10.2 # 10.32, 10.36, 10.40, 10.41, 10.46, 10.47, 10.49.
§10.3 # 10.71–73.
§10.5 # All.
§11.1 # 11.11–14, 11.16, 11.18, 11.22–31.
§11.2 # 11.44.
§11.3 # 11.55–70.
§11.4 # 11.97, 11.98, 11.100–103, 11.105–125.
Comments §10.5: Take nonsingular to mean injective/invertible, and singular to mean not injective/invertible. (Just as we call non-invertible matrices singular.)

Chapter 45 The matrix of a linear transformation

Problems §12.1 All except # 12.27, 12.33, 12.35, 12.36, 12.41–43.
§12.2 All except # 12.51–53.
§12.3 All except # 12.107.
§12.4 All except # 12.125–129, 12.139.

Chapter 46 Similarity of linear operators

Problems §13.2 All.
§13.3 # 13.66.
§13.4 All except # 13.88, 13.89.