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Lessons for Linear Algebra:
Play, Learn, Love Linear Algebra
Mark Fitch
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Front Matter
1
Basic Concepts
1.1
Solving Linear Systems
1.1.1
Solving by Elimination
1.1.2
A More Efficient Notation
1.1.3
Non-Unique Solutions
1.1.4
Terminology
1.2
Vector Arithmetic
1.2.1
Interpretation
1.2.2
Vector Arithmetic
1.2.3
Span
1.2.4
Matrix Arithmetic
1.3
Matrix Inverse
1.3.1
From Old to New
1.3.2
Properties of Matrix Inverses
1.3.3
Transposition
1.4
Dependence
1.4.1
Discovering a Relationship
1.4.2
Properties
1.5
Homogeneous
1.5.1
Discovering Non-Homogeneous Solution Forms
1.5.2
Proof
2
Connections and Functions
2.1
Big Theorem
2.1.1
Similar Problems
2.1.2
Big Theorem Begins
2.2
Arithmetic Theorems
2.2.1
Algebraic Properties
2.2.2
Proofs
2.3
Transformations
2.3.1
Experiment: Property
2.3.2
Linear Transformation Theorems
2.3.3
The Convenience of Being Linear
2.3.4
Matrix Notation
2.3.5
Big Theorem: Linear Transformations
3
Determinant
3.1
Determinant
3.1.1
Definition of Determinant
3.1.2
Efficient Algorithm
3.1.3
Additional Theorems
3.2
Cramer’s Rule
3.2.1
Cramer’s Rule Illustrated
3.2.2
Formula for Inverse
3.2.3
Computational Efficiency
4
Vector Spaces
4.1
Vector Spaces
4.1.1
Definition
4.1.2
Special Spaces
4.1.3
Finding Spaces
4.2
Basis
4.2.1
Independence
4.2.2
Span and Dependence
4.2.3
Basis
4.3
Coordinates
4.3.1
Definition of Coordinates
4.3.2
Working in Arbitrary Vector Spaces
4.3.3
Coordinates for Each Basis
4.4
Dimension
4.4.1
Uniqueness
4.4.2
Dimension
4.5
Rank
4.5.1
Row Spaces
4.5.2
Column Spaces
4.5.3
Null Spaces
4.5.4
Terminology
4.6
Change of Basis
4.6.1
Method for Changing Basis
4.6.2
Executing Change of Basis
4.6.3
Definition of Change of Basis Matrix
5
Eigenvectors
5.1
Eigen Vector
5.1.1
Motivation
5.1.2
Method
5.1.3
Independence of Eigenvectors
5.2
Diagonalization
5.2.1
Definitions
5.2.2
Motivation
5.2.3
Method
5.3
Linear Transformations and Diagonalization
5.3.1
Return to Matrix of a Transformation
6
Inner Product
6.1
Inner Product
6.1.1
Definition
6.1.2
Identifying Inner Products
6.1.3
Norm
6.1.4
Orthogonal
6.2
Orthogonal Sets
6.2.1
Orthogonal Sets
6.2.2
Orthogonality and Independence
6.2.3
Orthogonal Bases
6.2.4
Orthonormal
6.3
Orthogonal Projection
6.3.1
Projection
6.3.2
Projection onto a Space
6.3.3
Approximation Theorem
6.4
Gram-Schmidt
6.4.1
Review of Orthonormal Bases
6.4.2
Procedure
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Lessons for Linear Algebra:
Play, Learn, Love Linear Algebra
Mark Fitch
Department of Mathematics and Statistics
University of Alaska Anchorage
mafitch@alaska.edu
November 6, 2023
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