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Chapter 5: Vector Space Rⁿ

Chapter 5: Vector Space Rⁿ

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Chapter 5: Vector Space Rⁿ
5.1 Subspaces and Spanning
Subspaces of \mathbb { R } ^ { n }
Definition 5.1 Subspace of Rn
Example 5.1.1
Example 5.1.2
Solution.
Example 5.1.3
Spanning Sets
Definition 5.2 Linear Combinations and Span in Rn
Example 5.1.4
Theorem 5.1.1: Span Theorem
Proof.
Example 5.1.5
Example 5.1.6
Example 5.1.7
Example 5.1.8
Exercises for 5.1
Exercise 5.1.17
5.2 Independence and Dimension
Linear Independence
Definition 5.3 Linear Independence in Rn
Theorem 5.2.1
Independence Test
Example 5.2.1
Example 5.2.2
Example 5.2.3
Example 5.2.4
Example 5.2.5
Example 5.2.6
Example 5.2.7
Example 5.2.8
Theorem 5.2.2
Theorem 5.2.3
Example 5.2.9
Dimension
Theorem 5.2.4: Fundamental Theorem
Definition 5.4 Basis of Rn
Theorem 5.2.5: Invariance Theorem
Definition 5.5 Dimension of a Subspace of Rn
Example 5.2.10
Example 5.2.11
Example 5.2.12
Theorem 5.2.6
Example 5.2.13
Theorem 5.2.7
Theorem 5.2.8
Example 5.2.14
Proof.
Exercises for 5.2
5.3 Orthogonality
Dot Product, Length, and Distance
Definition 5.6 Length in Rn
Example 5.3.1
Theorem 5.3.1
Example 5.3.2
Example 5.3.3
Theorem 5.3.2: Cauchy Inequality9
Corollary 5.3.1: Triangle Inequality
Definition 5.7 Distance in Rn
Theorem 5.3.3
Orthogonal Sets and the Expansion Theorem
Definition 5.8 Orthogonal and Orthonormal Sets
Example 5.3.4
Example 5.3.5
Definition 5.9 Normalizing an Orthogonal Set
Example 5.3.6
Theorem 5.3.4: Pythagoras’ Theorem
Theorem 5.3.5
Theorem 5.3.6: Expansion Theorem
Example 5.3.7
Exercises for 5.3
Exercise 5.3.12
Exercise 5.3.13
Exercise 5.3.14
5.4 Rank of a Matrix
Definition 5.10 Column and Row Space of a Matrix
Lemma 5.4.1
Lemma 5.4.2
Example 5.4.1
Theorem 5.4.1: Rank Theorem
Example 5.4.2
Corollary 5.4.1
Corollary 5.4.2
Corollary 5.4.3
Lemma 5.4.3
Corollary 5.4.4
Theorem 5.4.2
Example 5.4.3
Theorem 5.4.3
Theorem 5.4.4
Example 5.4.4
Exercises for 5.4
Exercise 5.4.3
Exercise 5.4.15
5.5 Similarity and Diagonalization
Similar Matrices
Definition 5.11 Similar Matrices
Example 5.5.1
Definition 5.12 Trace of a Matrix
Lemma 5.5.1
Theorem 5.5.1
Example 5.5.2
Diagonalization Revisited
Theorem 5.5.2
Example 5.5.3
Theorem 5.5.3
Theorem 5.5.4
Theorem 5.5.5
Example 5.5.4
Lemma 5.5.2
Definition 5.13 Eigenspace of a Matrix
Lemma 5.5.3
Theorem 5.5.6
Example 5.5.5
Complex Eigenvalues
Example 5.5.6
Symmetric Matrices13
Theorem 5.5.7
Example 5.5.7
Exercises for 5.5
Exercise 5.5.13
5.6 Best Approximation and Least Squares
Definition 5.14 Normal Equations
Theorem 5.6.1: Best Approximation Theorem
Example 5.6.1
Example 5.6.2
Least Squares Approximation
Theorem 5.6.2
Example 5.6.3
Least Squares Approximating Polynomials
Definition 5.15 Least Squares Approximation
Theorem 5.6.3
Example 5.6.4
Other Functions
Theorem 5.6.4
Example 5.6.5
Exercises for 5.6
Exercise 5.6.10
5.7 An Application to Correlation and Variance
Theorem 5.7.1
Variance Formula
Correlation Formula
Data Scaling
Exercises for 5.7
Supplementary Exercises for Chapter 5
Section worksheets