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Linear Algebra Companion:
A supplement to undergraduate linear algebra enhanced by computation
Thomas R. Shemanske
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Front Matter
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Colophon
Preface
1
Matrices and basic operations
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1.1
Entering a matrix and row reduction
1.2
Solving systems of linear equations — mechanics
1.3
Solving systems of linear equations — theory
1.4
Exercises (with solutions)
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1.4.1
Linear equations — Mechanics
1.4.1
Exercises
1.4.2
Linear equations — Theory
1.4.2
Exercises
1.5
Generate your own matrices with prescribed properties
2
Vector Spaces, Subspaces and Linear Maps
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2.1
Vector spaces
2.2
Constructing Subspaces
2.3
Sums and Direct Sums
2.4
Viewing subspaces through different lenses
2.5
Linear maps and associated subspaces
2.6
Bases: the critical ingredient
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2.6
Exercises
2.7
Exercises (with solutions)
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2.7
Exercises
3
Constructing bases
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3.1
Linear dependence and independence
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3.1
Exercises
3.2
Constructing bases in
F
m
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3.2.1
An algorithmic approach
3.2.2
Recovering familiar results
3.3
Exercises (with solutions)
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3.3
Exercises
3.4
Using Sage to answer questions of independence and dependence
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3.4.1
Using Sage to check if a set of vectors in linearly independent
3.4.2
Using Sage to check if a vector is in the span of a set
3.4.3
Using Sage to understand the row and column space
4
Review of Core Topics
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4.1
Measuring injectivity and surjectivity
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4.1.1
Injective and surjective linear maps: assessment and implications.
4.1.2
Notions connected to isomorphism
4.2
Rank and Nullity
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4.2.1
Some fundamental subspaces
4.2.2
The rank-nullity theorem
4.2.3
Computing rank and nullity
4.2.4
Elementary Row and Column operations
4.2.4
Exercises
4.3
Coordinates and Matrices
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4.3.1
Coordinate Vectors
4.3.2
Matrix of a linear map
4.3.3
Matrix associated to a composition
4.3.4
Change of basis
4.4
Eigenvalues, eigenvectors, diagonalization
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4.4.1
An overview
4.4.2
Taking stock of where we are
4.4.3
An alternate characterization of diagonalizable
4.5
Minimal and characteristic polynomials
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4.5.1
Annihilating polynomials
4.5.2
The minimal polynomial
4.5.3
The characteristic polynomial
4.6
Exercises (with solutions)
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4.6
Exercises
4.7
Some Sage examples
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4.7.1
Eigenvalues, eigenvectors, and diagonalization
4.7.2
Rational and Jordan canonical forms
5
Inner Product Spaces
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5.1
Inner Product Spaces
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5.1.1
Definitions and examples
5.1.2
Basic Properties
5.2
Orthogonality and applications
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5.2.1
Orthogonal and Orthonormal Bases
5.2.2
Orthogonal complements and projections
5.2.3
What good is an orthogonal complement anyway?
5.2.4
Orthogonal Projections
5.2.5
A first look at the four fundamental subspaces
5.3
Orthogonal Projections and Least Squares Approximations
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5.3.1
Orthonormal bases and orthogonal/unitary matrices.
5.3.2
Sage Compuations
5.3.3
More on orthogonal projections
5.3.4
Least Squares Examples
5.4
Diagonalization of matrices in Inner Product Spaces
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5.4.1
Some relations between
A
and
A
∗
5.4.2
A closer look at matrices
A
∗
A
and
.
A
A
∗
.
5.5
Singular Value Decomposition
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5.5.1
SVD for linear maps
5.5.2
SVD for matrices
5.5.3
An algorithm for producing an SVD
5.5.4
Can an SVD for a matrix
A
be computed from
A
A
∗
instead?
5.5.5
Some Sage computations for an SVD
5.5.6
Deductions from seeing an SVD
5.5.7
SVD and image processing
5.5.8
Some parting observations on the SVD
5.6
Exercises (with solutions)
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5.6
Exercises
6
Basic Definitions and Examples
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6.1
Definitions
6.2
Some familiar examples of vector spaces
Back Matter
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References and Suggested Readings
Linear Algebra Companion:
A supplement to undergraduate linear algebra enhanced by computation
Thomas R. Shemanske
Department of Mathematics
Dartmouth College
thomas.r.shemanske@dartmouth.edu
Version: January 10, 2025
Colophon
Preface
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