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Linear Algebra Refresher:
Review, Amplification, Examples
Thomas R. Shemanske
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Front Matter
Colophon
Preface
1
A quick review of a first course
1.1
Vector spaces and linear maps
1.1.1
Some familiar examples of vector spaces
1.1.2
Linear independent and spanning sets
1.1.2
Exercises
1.1.3
Defining a linear map.
1.1.3
Exercises
1.2
Measuring injectivity and surjectivity
1.2.1
Injective and surjective linear maps: assessment and implications.
1.2.2
Notions connected to isomorphism
1.3
Rank and Nullity
1.3.1
Some fundamental subspaces
1.3.2
The rank-nullity theorem
1.3.3
Computing rank and nullity
1.3.4
Elementary Row and Column operations
1.3.4
Exercises
1.4
Coordinates and Matrices
1.4.1
Coordinate Vectors
1.4.2
Matrix of a linear map
1.4.3
Matrix associated to a composition
1.4.4
Change of basis
1.5
Eigenvalues, eigenvectors, diagonalization
1.5.1
The big picture
1.5.2
Taking stock of where we are
1.5.3
An alternate characterization of diagonalizable
1.6
Minimal and characteristic polynomials
1.6.1
Annihilating polynomials
1.6.2
The minimal polynomial
1.6.3
The characteristic polynomial
1.7
Some Sage examples
1.7.1
Row reduction, echelon form, kernel, column space
1.7.2
Eigenvalues, eigenvectors, and diagonalization
1.7.3
Rational and Jordan canonical forms
1.8
Exercises (with solutions)
1.8
Exercises
2
Vector space constructions
2.1
Sums and Direct Sums
2.2
Quotient Spaces
2.3
Linear Maps out of quotients
3
Inner Product Spaces
3.1
Inner Product Spaces
3.1.1
Definitions and examples
3.1.2
Basic Properties
3.2
Orthogonality and applications
3.2.1
Orthogonal and Orthonormal Bases
3.2.2
Orthogonal complements and projections
3.2.3
What good is an orthogonal complement anyway?
3.2.4
Orthogonal Projections
3.2.5
A first look at the four fundamental subspaces
3.3
Orthogonal Projections and Least Squares Approximations
3.3.1
Orthonormal bases and orthogonal/unitary matrices.
3.3.2
Sage Compuations
3.3.3
More on orthogonal projections
3.3.4
Least Squares Examples
3.4
Diagonalization of matrices in Inner Product Spaces
3.4.1
Some relations between
\(A\)
and
\(A^*\)
3.4.2
A closer look at matrices
\(A^*A\)
and
\(AA^*\text{.}\)
3.5
Adjoint Maps and properties
3.5.1
Basic Properties
3.5.2
A second look at the four fundamental subspaces
3.6
Singular Value Decomposition
3.6.1
SVD for linear maps
3.6.2
SVD for matrices
3.6.3
An algorithm for producing an SVD
3.6.4
Can an SVD for a matrix
\(A\)
be computed from
\(AA^*\)
instead?
3.6.5
Some Sage computations for an SVD
3.6.6
Deductions from seeing an SVD
3.6.7
SVD and image processing
3.6.8
Some parting observations on the SVD
3.7
Exercises (with solutions)
3.7
Exercises
4
Definitions and Examples
4.1
Basic Definitions and Examples
Back Matter
References and Suggested Readings
Linear Algebra Refresher:
Review, Amplification, Examples
Thomas R. Shemanske
Department of Mathematics
Dartmouth College
thomas.r.shemanske@dartmouth.edu
Version: January 19, 2025
Colophon
Preface
