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Linear Algebra Refresher:
Review, Amplification, Examples
Thomas R. Shemanske
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Front Matter
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Colophon
Preface
1
A quick review of a first course
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1.1
Vector spaces and linear maps
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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
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1.2.1
Injective and surjective linear maps: assessment and implications.
1.2.2
Notions connected to isomorphism
1.3
Rank and Nullity
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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
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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
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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
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1.6.1
Annihilating polynomials
1.6.2
The minimal polynomial
1.6.3
The characteristic polynomial
1.7
Some Sage examples
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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)
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1.8
Exercises
2
Vector space constructions
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2.1
Sums and Direct Sums
2.2
Quotient Spaces
2.3
Linear Maps out of quotients
3
Inner Product Spaces
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3.1
Inner Product Spaces
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3.1.1
Definitions and examples
3.1.2
Basic Properties
3.2
Orthogonality and applications
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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
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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
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3.4.1
Some relations between
A
and
A
∗
3.4.2
A closer look at matrices
A
∗
A
and
.
A
A
∗
.
3.5
Adjoint Maps and properties
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3.5.1
Basic Properties
3.5.2
A second look at the four fundamental subspaces
3.6
Singular Value Decomposition
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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
A
A
∗
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)
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3.7
Exercises
4
Definitions and Examples
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4.1
Basic Definitions and Examples
Back Matter
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References and Suggested Readings
Chapter
4
Definitions and Examples
Here we accumulate basic definitions and examples from a standard first course in linear algebra.
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4.1
Basic Definitions and Examples
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