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Linear Algebra:
Companion notes
Abhay Soman
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\(\newcommand{\R}{\mathbb R} \def\C{\mathbb{C}} \def\Q{\mathbb{Q}} \def\N{\mathbb{N}} \def\Z{\mathbb{Z}} \def\ev{\mathrm{ev}} \def\Im{\mathrm{Im}} \def\End{\mathrm{End}} \def\Aut{\mathrm{Aut}} \def\Hom{\mathrm{Hom}} \def\Iso{\mathrm{Iso}} \def\GL{\mathrm{GL}} \def\ker{\mathrm{ker }} \def\Span{\mathrm{Span}} \def\tr{\mathrm{tr}} \newcommand{\unit}{1\!\!1} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Preface
1
Elementary row operations
1.1
Elementary row operations
1.2
Exercises
1.3
Row echelon form using SageMath
1.4
Uniqueness of row reduced echelon form
2
Vector spaces
2.1
Definition of a vector space
2.2
Examples of vector spaces
2.3
Exercises
2.4
Linear combination
2.5
Subspace of a vector space
2.6
Exercises
3
Basis and dimension of a vector space
3.1
Linearly independent vectors
3.2
Exercises
3.3
Basis and Dimension
3.4
Invariance of dimension
3.5
Exercises
3.6
Examples
3.7
Sum and direct sum of vector subspaces
4
Linear transformations
4.1
Definition of linear transformation
4.2
Examples of linear transformations
4.3
Exercises
4.4
Kernel and image of a linear homomorphism
4.5
Rank-Nullity Theorem
4.6
A basis for the kernel of trace map
4.7
Ordered basis and linear maps
4.7.1
Ordered basis
4.7.2
Matrix representation of a linear transformation
4.8
Exercises
4.9
Exact sequences (Optional)
4.10
Exercises
5
Isomorphisms
5.1
Definition of Isomorphism
5.2
Matrices and a space of linear transformations
5.3
Isomorphism and invertible matrix
5.4
Row and Column rank
5.5
Rank and Nullity using SageMath
5.6
Natural isomorphism between a vector space and its double dual
5.7
Exercises
6
Quotient space
6.1
Definition of Quotient Space
6.2
Natural Projection and Correspondence Theorem
6.3
Fundamental Homomorphism Theorem
6.4
Examples
7
Invariant Subspaces and Eigenvectors
7.1
Invariant subspaces
7.2
Eigenvalues and Eigenvectors
7.3
Algebraic and Geometric multiplicity
7.4
Triangulable linear maps and matrices
7.5
Cyclic subspaces
7.6
Some Computations with Jordan block
7.7
Exercises
7.8
Eigenvalues and Eigenvectors using SageMath
8
Cayley-Hamilton theorem and Jordan Normal Form
8.1
Cayley-Hamilton Theorem
8.2
Finding Jordan normal form over
\(\C\)
8.3
Examples
Backmatter
A
(Algebra of polynomials and determinants)
A.1
Algebra of polynomials
A.2
Determinants
Colophon
Chapter
4
Linear transformations
In this chapter, we study linear transformations. These are maps between vector spaces that ‘preserve’ vector space operations. We will also see a relation between linear transformations and matrices.
4.1
Definition of linear transformation
4.2
Examples of linear transformations
4.3
Exercises
4.4
Kernel and image of a linear homomorphism
4.5
Rank-Nullity Theorem
4.6
A basis for the kernel of trace map
4.7
Ordered basis and linear maps
4.8
Exercises
4.9
Exact sequences (Optional)
4.10
Exercises
