# :: Practise Problems for Discover Linear Algebra (One Semester Version)

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

## Chapter 1 Systems of linear equations

 Problems §§3.1-3 All exercises. §3.5 # 3.53-58, 3.64-66, 3.70, 3.71. §2.5 # 2.80-83, 2.88 Comment The supplementary text sometimes uses different terminology from what Discover Linear Algebra uses. For now, take the phrase "vector in $$\mathbf{R}^n$$" to mean an "ordered list of $$n$$ values." For example, the "vector" $$(2,3,5)$$ could correspond to variable values $$x = 2$$, $$y = 3$$, $$z = 5$$ in a system that involves variables $$x,y,z$$, or it could correspond to variable values $$x_1 = 2$$, $$x_2 = 3$$, $$x_3 = 5$$ in a system that involves variables $$x_1,x_2,x_3$$.

## Chapter 2 Solving systems using matrices

 Problems §2.6 # 2.92-94, 2.96-2.99. §3.6 # 3.74-6, 3.78-80, 3.84, 3.85(b), 3.86(b), 3.87. §2.7 # 2.104-106, 2.108-110, 2.112-115. §3.7 # 3.90-100, 3.102-105. §3.8 # 3.110-117. §3.9 # 3.124, 3.129, 3.130. Comments Terminology: The supplementary text uses the phrase "echelon form" to mean almost the same thing as defined in Discover Linear Algebra; the difference is that their version of "echelon form" does not require the leading entries to be 1s. See 2.94 in 3000 Solved Problems. Terminology: The supplementary text uses the phrase "row canonical form" to mean the same thing as the phrase "reduced row echelon form" in Discover Linear Algebra. See 2.104 in 3000 Solved Problems. §3.7: Convert these to augmented matrices and solve by row reducing.

## Chapter 4 Matrices and matrix operations

 Problems §2.1-4 All exercises, BUT omit # 2.36, 2.52, 2.55-58. §3.8 # 3.108, 3.109. §3.9 # 3.121, 3.122. §3.10 # 3.133, 3.135. §4.4 # 4.45, 4.46, 4.49, 4.50, 4.53, 4.56, 4.57, 4.59. Comments §2.2: In Theorem 2.1 on page 31 of the supplementary text, and in the related problems following it, take the "field of scalars K" to mean the "field of real numbers $$\mathbb{R}$$." §4.5 # 4.53: Matrix $$B$$ is provided at the bottom of the previous page. §4.5 # 4.59: Look for the pattern in # 4.56,4.57. Maybe compute $$A^4$$ to confirm the pattern.

## Chapter 5 Matrix inverses

 Problems §4.2 # 4.13, 4.14. §4.6 # 4.80, 4.82–84, 4.85, 4.88–91, 4.97, 4.99–104, 4.106, 4.107.

## Chapter 6 Elementary matrices

 Problems §3.5 # 3.59–61. §2.5 # 2.79, 2.84. §4.7 # 4.108–112, 4.116, 4.117, 4.122, 4.123.

## Chapter 7 Special forms of square matrices

 Problems §4.1 # 4.1, 4.2, 4.4, 4.19 (just read), 4.20–31. §4.9 # 4.140–143, 4.150, 4.151, 4.153, 4.154, 4.157–159, 4.161. §4.6 # 4.105. §4.10 # 4.165, 4.166 (just read), 4.168–180, 4.184–186, 4.189, 4.190 (just $$A^2$$), 4.192. Comments §4.10 # 4.166: Skew-symmetric matrices are not explored in Discover Linear Algebra, but you can read what they are in problem 4.166 of 3000 Solved Problems, and then work on some related problems following 4.166.

## Chapter 8 Determinants

 Problems §5.1 # 5.4–6. §5.2 # 5.7–17. §5.5 # 5.88. §5.6 # 5.90–99. §5.3 # 5.31–36.

## Chapter 9 Determinants versus row operations

 Problems §5.5 # 5.89. §5.6 # 5.102–109, 5.111. §5.11 # 5.166.

## Chapter 10 Determinants, the adjoint, and inverses

 Problems §5.3 # 5.40–42. §5.5 # 5.77, 5.80–86. §5.7 # 5.114–117, 5.119–135. §5.2 # 5.19–22. §5.9 # 5.145–147. Comments Solve by determinants: Any question that says "solve by determinants" or "express the solution in terms of determinants" intends for you to use Cramer's Rule (see Subsection 10.3.8 and Subsection 10.5.4.)

## Chapter 11 Introduction to vectors

 Problems §1.1 All exercises. §1.2 # 1.12–25. §1.8 # 1.98, 1.99. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra.

## Chapter 12 Geometry of vectors

 Problems §1.4 # 1.46–51, 1.55. §1.5 # 1.60–68, 1.73, 1.74. §1.6 # 1.76–78, 1.80, 1.81. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. §1.6 # 1.76–78: The notation $$d(u,v)$$ means "the distance between (the heads of) vectors $$u$$ and $$v$$."

## Chapter 13 Orthogonal vectors

 Problems §1.6 # 1.83, 1.84. §1.7 # 1.86–90, 1.92, 1.93, 1.95. §1.8 # 1.100, 1.101, 1.103–105. §1.12 # 1.161–170, 1.172–176. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. §1.6: The supplementary textbook uses the notation $$\operatorname{proj}(u,v)$$ to mean "the projection of vector u onto vector v." In Discover Linear Algebra, the symbols $$\operatorname{proj}_{\mathbf{v}}\mathbf{u}$$ are used.

## Chapter 14 Geometry of linear systems

 Problems §1.8 # 1.108–110, 1.112. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra.

## Chapter 15 Abstract vector spaces

 Problems §7.1 # 7.2–6, 7.8–12, 7.17–29, 7.32. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. The symbol $${}\in{}$$ means "...is an object in the collection...." For example, the string of symbols $$u,v \in V$$ can be read as "$$u$$ and $$v$$ are objects in the collection $$V$$." §7.1: Replace any mention of "the field of scalars $$K$$" by the collection $$\mathbb{R}$$ of real numbers (which act as scalars for us). Also, replace any occurrence of $$K^n$$ by the collection $$\mathbb{R}^n$$ of $$n$$-dimensional vectors. The supplementary textbook provides only eight vector space axioms (in problem #7.1), but the other two axioms are hiding in the preamble. §7.1 # 7.2: You should attempt to prove these from the axioms and ignore the provided solution. Proofs of these statements appear in Discover Linear Algebra — see Section 15.6: Theory.

## Chapter 16 Subspaces

 Problems §7.2 # 7.38, 7.41–55, 7.58, 7.61–63. §7.3 # 7.66–69, 7.70, 7.72–74, 7.76–80, 7.86. §7.4 # 7.89–99. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. The symbol $${}\in{}$$ means "...is an object in the collection...." For example, the string of symbols $$u,v \in V$$ can be read as "$$u$$ and $$v$$ are objects in the collection $$V$$." The symbol $${}\subseteq{}$$ means "...is a subcollection of...." For example, if $$W$$ and $$V$$ are collections of vectors, writing $$W \subseteq V$$ means that every vector in the collection $$W$$ is also part of the collection $$V$$. §7: Replace any mention of "the field of scalars $$K$$" by the collection $$\mathbb{R}$$ of real numbers (which act as scalars for us). Also, replace any occurrence of $$K^n$$ by the collection $$\mathbb{R}^n$$ of $$n$$-dimensional vectors. §7.4 # 7.99: The assumption about $$u_k$$ is for one particular value of $$k \gt 1$$, not each value of $$k \gt 1$$.

## Chapter 17 Linear independence

 Problems §8.1 # 8.2, 8.3, 8.6–11. §8.2 # 8.13–26. §8.3 # 8.32–34, 8.40. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. The symbol $${}\in{}$$ means "...is an object in the collection...." For example, the string of symbols $$u,v \in V$$ can be read as "$$u$$ and $$v$$ are objects in the collection $$V$$." §8: Replace any mention of "the field of scalars $$K$$" by the collection $$\mathbb{R}$$ of real numbers (which act as scalars for us). Also, replace any occurrence of $$K^n$$ by the collection $$\mathbb{R}^n$$ of $$n$$-dimensional vectors. The supplementary text uses what is called the Test for Linear Dependence/Independence in Discover Linear Algebra as the definition of dependence/independence. But, as explored in Discovery 17.3, the test and definition are equivalent.

## Chapter 18 Basis and Coordinates

 Problems §8.3 # 8.29, 8.38. §8.4 # 8.46, 8.53–59. §8.5 # 8.79. §8.9 # 8.128–141, 8.143, 8.145–151. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. The symbol $${}\in{}$$ means "...is an object in the collection...." For example, the string of symbols $$u,v \in V$$ can be read as "$$u$$ and $$v$$ are objects in the collection $$V$$." The symbol $${}\subseteq{}$$ means "...is a subcollection of...." For example, if $$W$$ and $$V$$ are collections of vectors, writing $$W \subseteq V$$ means that every vector in the collection $$W$$ is also part of the collection $$V$$. §8: Replace any mention of "the field of scalars $$K$$" by the collection $$\mathbb{R}$$ of real numbers (which act as scalars for us). Also, replace any occurrence of $$K^n$$ by the collection $$\mathbb{R}^n$$ of $$n$$-dimensional vectors. §8.3 # 8.38: See the definition in # 8.37. §8.4: In questions where you are to determine whether some set of vectors is a basis, do not use dimension arguments — check directly whether it is both linearly independent and a spanning set. §8.5 # 8.79: The provided Method 2 solution will make more sense after you learn about column and row spaces of a matrix in Chapter 20: Column, row, and null spaces. You can check independence by row reducing a matrix whose columns are the given vectors.

## Chapter 19 Dimension

 Problems §8.3 # 8.36, 8.39. §8.4 # 8.47–49, 8.52, 8.57, 8.60–64. §8.5 # 8.69–78, 8.80–84. §8.7 # 8.102–104. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. The symbol $${}\in{}$$ means "...is an object in the collection...." For example, the string of symbols $$u,v \in V$$ can be read as "$$u$$ and $$v$$ are objects in the collection $$V$$." The symbol $${}\subseteq{}$$ means "...is a subcollection of...." For example, if $$W$$ and $$V$$ are collections of vectors, writing $$W \subseteq V$$ means that every vector in the collection $$W$$ is also part of the collection $$V$$. §8: Replace any mention of "the field of scalars $$K$$" by the collection $$\mathbb{R}$$ of real numbers (which act as scalars for us). Also, replace any occurrence of $$K^n$$ by the collection $$\mathbb{R}^n$$ of $$n$$-dimensional vectors. §8.5 # 8.72, 8.77, 8.78, 8.81, 8.84: The provided solutions will make more sense after you learn about row space of a matrix in Chapter 20: Column, row, and null spaces. For now, you can check independence by row reducing a matrix whose columns are the given vectors. §8.7 # 8.104: The symbol $${}\cap{}$$ means "intersection of...." So, in this question, $$U \cap W$$ means "the collection of all vectors that are simultaneously part of both collections, $$U$$ and $$W$$." An intersection of subspaces is always a subspace, because they share the zero vector (and so are non-empty), and they are both closed under the vector operations (see problem # 7.59).

## Chapter 20 Column, row, and null spaces

### Column space

The supplementary textbook does not have many problems on column space. Instead, some practise problems are provided in the following pages.

### Row space

 Problems §7.5 # 7.102–109. §8.3 # 8.32. §8.5 # 8.72, 8.77, 8.78. (See comments below.) Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. The symbol $${}\in{}$$ means "...is an object in the collection...." For example, the string of symbols $$u,v \in V$$ can be read as "$$u$$ and $$v$$ are objects in the collection $$V$$." The symbol $${}\subseteq{}$$ means "...is a subcollection of...." For example, if $$W$$ and $$V$$ are collections of vectors, writing $$W \subseteq V$$ means that every vector in the collection $$W$$ is also part of the collection $$V$$. §8: Replace any mention of "the field of scalars $$K$$" by the collection $$\mathbb{R}$$ of real numbers (which act as scalars for us). Also, replace any occurrence of $$K^n$$ by the collection $$\mathbb{R}^n$$ of $$n$$-dimensional vectors. §8.5 # 8.72, 8.77, 8.78: Some of these appear in previous lists above, but you were advised to work with the vectors as columns. (So, effectively, they became column space questions.) You can now repeat as row space questions: use the vectors as rows in a matrix, and row reduce to determine a basis for the row space of the matrix. (Note that this will give you a different answer than when answering as a column space question.)

### Null space

 Problems §8.7 # 8.96–101. §3.9 # 3.123, 3.125–128. Comments §§8.7,3.9: The solutions given here are by reducing systems of equations using "equation operations," but you can (and should) reduce the system using a matrix and row operations instead.

## Chapter 21 Eigenvalues and eigenvectors

NOTE   Students may wish to save their work for this set of practise problems, because in the set of practise problems for Chapter 22 are listed some of the problems that are being skipped over here and which follow up on some of these problems.

 Problems §16.1 # 16.1–4, 16.13, 16.14, 16.17. §16.2 # 16.26, 16.28, 16.30, 16.31, 16.35, 16.40. §16.3 # 16.42–45, 16.47, 16.49, 16.51, 16.52, 16.56–58, 16.60, 16.62, 16.63, 16.65, 16.72–74. Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. The supplementary textbook uses the variable $$t$$ instead of $$\lambda$$ for computing eigenvalues, and uses the notation $$\Delta t$$ to mean the characteristic polynomial of a matrix. §16: Replace any mention of "the field of scalars $$K$$" by the collection $$\mathbb{R}$$ of real numbers (which act as scalars for us). Also, replace any occurrence of $$K^n$$ by the collection $$\mathbb{R}^n$$ of $$n$$-dimensional vectors. §16.2 # 16.26: Ignore the first sentence. Carry out the instructions in the second and third sentence for $$I$$ (the identity matrix). §16.2 # 16.30: Think of $$T$$ as a square matrix, and $$T(v)$$ to mean matrix $$T$$ times vector $$v$$. §16.2 # 16.31: Consider $$T$$ as a square matrix. (That is, ignore the first sentence of Theorem 16.4). In the second sentence of the theorem, replace the term operator by the term square matrix, and in the third sentence of the theorem, replace the term kernelkernel by the term null space. §16.2 # 16.40: Same comment as for #16.30 above. §16.3: In any question that contains instructions "find a maximum set of linearly independent eigenvectors of the matrix," instead change the instructions to "determine a basis for each eigenspace of the matrix."

## Chapter 22 Diagonalization

NOTE   A number of these questions are follow-ups to problems listed above under Chapter 21. In §16.3 in particular, when a question refers to work from a problem in that previous list, this is indicated in the list below by "(see ...)" after the question.

 Problems §16.1 # 16.21. §16.2 # 16.34, 16.36, 16.38. §16.3 # 16.46 (see 16.42–45), 16.48 (see 16.47), 16.50 (see 16.49), 16.51, 16.53 (see 16.52), 16.59 (see 16.56–58), 16.61 (see 16.56–60), 16.64 (see 16.62,63), 16.66 (see 16.62–64), 16.67 (see 16.62–66), 16.68–71, 16.75–76 (see 16.72–74). Comments Notation: The supplementary textbook does not use bold letters to represent vectors as is done in Discover Linear Algebra. The supplementary textbook uses the variable $$t$$ instead of $$\lambda$$ for computing eigenvalues, and uses the notation $$\Delta t$$ to mean the characteristic polynomial of a matrix. §16: Replace any mention of "the field of scalars $$K$$" by the collection $$\mathbb{R}$$ of real numbers (which act as scalars for us). Also, replace any occurrence of $$K^n$$ by the collection $$\mathbb{R}^n$$ of $$n$$-dimensional vectors.