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Chapter 6: Vector Spaces

Chapter 6: Vector Spaces

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Chapter 6: Vector Spaces
6.1 Examples and Basic Properties
Definition 6.1 Vector Spaces
Axioms for vector addition
Axioms for scalar multiplication
Example 6.1.1
Example 6.1.2
Example 6.1.3
Example 6.1.4
Example 6.1.5
Example 6.1.6
Example 6.1.7
Theorem 6.1.1: Cancellation
Theorem 6.1.2
Theorem 6.1.3
Proof.
Example 6.1.8
Example 6.1.9
Exercises for 6.1
Exercise 6.1.12
6.2 Subspaces and Spanning Sets
Definition 6.2 Subspaces of a Vector Space
Theorem 6.2.1: Subspace Test
Example 6.2.1
Example 6.2.2
Example 6.2.3
Example 6.2.4
Example 6.2.5
Example 6.2.6
Linear Combinations and Spanning Sets
Definition 6.3 Linear Combinations and Spanning
Example 6.2.7
Example 6.2.8
Example 6.2.9
Theorem 6.2.2
Example 6.2.10
Example 6.2.11
Exercises for 6.2
6.3 Linear Independence and Dimension
Definition 6.4 Linear Independence and Dependence
Example 6.3.1
Example 6.3.2
Example 6.3.3
Example 6.3.4
Example 6.3.5
Example 6.3.6
Solution.
Theorem 6.3.1
Theorem 6.3.2: Fundamental Theorem
Definition 6.5 Basis of a Vector Space
Theorem 6.3.3: Invariance Theorem
Definition 6.6 Dimension of a Vector Space
Example 6.3.7
Example 6.3.8
Example 6.3.9
Example 6.3.10
Example 6.3.11
Example 6.3.12
Exercises for 6.3
Exercise 6.3.10
Exercise 6.3.11
Exercise 6.3.29
6.4 Finite Dimensional Spaces
Lemma 6.4.1: Independent Lemma
Definition 6.7 Finite Dimensional and Infinite Dimensional Vector Spaces
Lemma 6.4.2
Theorem 6.4.1
Proof.
Example 6.4.1
Example 6.4.2
Example 6.4.3
Example 6.4.4
Theorem 6.4.2
Example 6.4.5
Lemma 6.4.3: Dependent Lemma
Theorem 6.4.3
Example 6.4.6
Theorem 6.4.4
Example 6.4.7
Example 6.4.8
Example 6.4.9
Theorem 6.4.5
Exercises for 6.4
Exercise 6.4.4
Exercise 6.4.6
Exercise 6.4.8
Exercise 6.4.22
6.5 An Application to Polynomials
Theorem 6.5.1
Corollary 6.5.1
Corollary 6.5.2
Remainder Theorem
Factor Theorem
Corollary 6.5.3: Taylor’s Theorem
Example 6.5.1
Theorem 6.5.2
Then
Proof.
Example 6.5.2
Theorem 6.5.3: Lagrange Interpolation Expansion
Example 6.5.3
Theorem 6.5.4
Exercises for 6.5
6.6 An Application to Differential Equations
Theorem 6.6.1
Theorem 6.6.2
Remark
Theorem 6.6.3
Example 6.6.1
Lemma 6.6.1
Theorem 6.6.4
Example 6.6.2
Theorem 6.6.5
Example 6.6.3
Theorem 6.6.6
Example 6.6.4
Exercises for 6.6
Exercise 6.6.4
Supplementary Exercises for Chapter 6
Section worksheets