4.1-4.6 Key Exercises

Math 2331 – Linear Algebra4.1-4.6 Vector Spaces

Key Exercises

Jiwen He

Department of Mathematics, University of Houston

jiwenhe@math.uh.edumath.uh.edu/∼jiwenhe/math2331

Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

4.1 Vector Spaces & Subspaces

Key Exercises 1–18, 23–24

Theorem 1 provides the main homework tool in this sectionfor showing that a set is a subspace.

Key Exercises: 1–18, 23–24.

Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 5 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 8 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 9 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 11 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 12 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 13 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 14 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Mark each statement True or False. Justify each answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 15 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Mark each statement True or False. Justify each answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 16 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

4.2 Null Spaces, Column Spaces, & LinearTransformations

Key Exercises 3–6, 17–26

This section provides a review of Chapter 1 using the newterminology.

Key Exercises: 3–6, 17–26. They are simple but helpful

Jiwen He, University of Houston Math 2331, Linear Algebra 17 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 18 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 19 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 20 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

A is an m × n matrix. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 21 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

A is an m × n matrix. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 22 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

4.3 Linearly Independent Sets; Bases

Key Exercises 21–25

The materials on bases for Null A and Col A are essential forSection 4.5 and 4.6.

Key Exercises: 21–25.

Jiwen He, University of Houston Math 2331, Linear Algebra 23 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Mark each statement True or False. Justify each answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 24 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Mark each statement True or False. Justify each answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 25 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 26 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 27 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 28 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

4.4 Coordinate Systems

Key Exercises 15–16, 25–26,32

Key Exercises: 15–16, 25–26,32.

Jiwen He, University of Houston Math 2331, Linear Algebra 29 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 30 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 31 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 32 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 33 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

4.5 The Dimension of a Vector Space

Key Exercises 19–22, 25–26,30

Theorem 9 is true because a vector space isomorphic to Rn

has the same algebraic properties as Rn.

Key Exercises: 19–22, 25–26,30

Jiwen He, University of Houston Math 2331, Linear Algebra 34 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

V is a vector space. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 35 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

V is a vector space. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 36 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 37 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 38 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 39 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 40 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 41 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

4.6 Rank

Key Exercises 17–30

Key Exercises: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 42 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

A is an m × n matrix. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 43 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

A is an m × n matrix. Mark each statement True or False. Justifyeach answer.

Jiwen He, University of Houston Math 2331, Linear Algebra 44 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 45 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 46 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 47 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 48 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 49 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 50 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 51 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 52 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 53 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 54 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 55 / 56

4.1-4.6 Key Exercises 4.1: 1–24 4.2: 3–26 4.3: 21–25 4.4: 15–32 4.5: 19–30 4.6: 17–30

Jiwen He, University of Houston Math 2331, Linear Algebra 56 / 56