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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
Colophon
Preface
1
Matrices and basic operations
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)
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
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
2.6
Exercises
2.7
Exercises (with solutions)
2.7
Exercises
3
Constructing bases
3.1
Linear dependence and independence
3.1
Exercises
3.2
Constructing bases in
\(F^m\)
3.2.1
An algorithmic approach
3.2.2
Recovering familiar results
3.3
Exercises (with solutions)
3.3
Exercises
3.4
Using Sage to answer questions of independence and dependence
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
4.1
Measuring injectivity and surjectivity
4.1.1
Injective and surjective linear maps: assessment and implications.
4.1.2
Notions connected to isomorphism
4.2
Rank and Nullity
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
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
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
4.5.1
Annihilating polynomials
4.5.2
The minimal polynomial
4.5.3
The characteristic polynomial
4.6
Exercises (with solutions)
4.6
Exercises
4.7
Some Sage examples
4.7.1
Eigenvalues, eigenvectors, and diagonalization
4.7.2
Rational and Jordan canonical forms
5
Inner Product Spaces
5.1
Inner Product Spaces
5.1.1
Definitions and examples
5.1.2
Basic Properties
5.2
Orthogonality and applications
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
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
5.4.1
Some relations between
\(A\)
and
\(A^*\)
5.4.2
A closer look at matrices
\(A^*A\)
and
\(AA^*\text{.}\)
5.5
Singular Value Decomposition
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
\(AA^*\)
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)
5.6
Exercises
6
Basic Definitions and Examples
6.1
Definitions
6.2
Some familiar examples of vector spaces
Back Matter
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 19, 2025
Colophon
Preface
