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Linear Algebra:
Companion notes
A. Soman
Contents
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Contents
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Front Matter
Colophon
Preface
1
Elementary row operations
Elementary row operations
Exercises
Row echelon form using SageMath
Uniqueness of row reduced echelon form
2
Vector spaces
Definition of a vector space
Examples of vector spaces
Exercises
Linear combination
Subspace of a vector space
Exercises
3
Basis and dimension of a vector space
Linearly independent vectors
Exercises
Basis and Dimension
Invariance of dimension
Exercises
Examples
Sum and direct sum of vector subspaces
4
Linear transformations
Definition of linear transformation
Examples of linear transformations
Exercises
Kernel and image of a linear homomorphism
Rank-Nullity Theorem
A basis for the kernel of trace map
Ordered basis and linear maps
Exercises
Exact sequences (Optional)
Exercises
5
Isomorphisms
Definition of Isomorphism
Matrices and a space of linear transformations
Isomorphism and invertible matrix
Row and Column rank
Rank and Nullity using SageMath
Natural isomorphism between a vector space and its double dual
Exercises
6
Quotient space
Definition of Quotient Space
Natural Projection and Correspondence Theorem
Fundamental Homomorphism Theorem
Examples
7
Invariant Subspaces and Eigenvectors
Invariant subspaces
Eigenvalues and Eigenvectors
Algebraic and Geometric multiplicity
Triangulable linear maps and matrices
Cyclic subspaces
Some Computations with Jordan block
Exercises
Eigenvalues and Eigenvectors using SageMath
8
Cayley-Hamilton theorem and Jordan Normal Form
Cayley-Hamilton Theorem
Finding Jordan normal form over \(\C\)
Examples
Back Matter
A
(Algebra of polynomials and determinants)
Algebra of polynomials
Determinants
Authored in PreTeXt
Linear Algebra:
Companion notes
A. Soman
School of Mathematics and Statistics
University of Hyderabad
Colophon
Preface