Topic

No. of hours

Lecture

Tutorial/

Practical

Basic Linear Algebra :-

Vector Spaces,

Subspaces,

Direct Sums,

Spanning Sets and Linear Independence,

The Dimension of a Vector Space,

Ordered Bases and Coordinate Matrices,

The Row and Column Spaces of a Matrix,

The Complexification of a Real Vector Space.

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Linear Transformations:

The Kernel and Image of a Linear Transformation,

Isomorphisms,

The Rank Plus Nullity Theorem,

Change of Basis Matrices,

The Matrix of a Linear Transformation,

Change of Bases for Linear Transformations,

Equivalence of Matrices,

Similarity of Matrices,

Similarity of Operators,

Invariant Subspaces and Reducing Pairs,

Projection Operators

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The Isomorphism Theorems :-

Quotient Spaces,

The Universal Property of Quotients and

the First Isomorphism Theorem,

Quotient Spaces, Complements and Codimension,

Additional Isomorphism Theorems,

Linear Functional,

Dual Bases,

Reflexivity,

Annihilators,

Operator Adjoints.

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The Structure of a Linear Operator :-

The Module Associated with a Linear Operator,

The Primary Cyclic Decomposition

The Characteristic Polynomial,

Cyclic and Indecomposable Modules,

The Big Picture,

The Rational Canonical Form.

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Eigenvalues and Eigenvectors ;-

Geometric and Algebraic Multiplicities,

The Jordan Canonical Form,

Triangularizability and Schur's Theorem,

Diagonalizable Operators.

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Real and Complex Inner Product Spaces,

Norm and Distance,

Isometries,

Orthogonality,

Orthogonal and Orthonormal Sets,

The Projection Theorem and Best Approximations,

The Riesz RepresentationTheorem,

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Structure Theory for Normal Operators :-

The Adjoint of a Linear Operator,

Orthogonal Projections

Unitary Diagonalizability,

Normal Operators,

Special Types of Normal Operators,

Self-Adjoint Operators,

Unitary Operators and Isometries,

The Structure of Normal Operators,

Functional Calculus,

Positive Operators,

The Polar Decomposition of an Operator.

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The Theory of Bilinear Forms :-

Symmetric, Skew-Symmetric and Alternate Forms.

The Matrix of a Bilinear Form.

Quadratic Forms,

Orthogonality,

Linear Functionals,

Orthogonal Complements and Orthogonal Direct Sums,

Isometries,

Hyperbolic Spaces,

Nonsingular Completions of a Subspace,

The Witt Theorems: A Preview,

The Classification Problem for Metric Vector Spaces.

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Symplectic Geometry,

The Structure of Orthogonal Geometries: Orthogonal Bases,

The Classification of Orthogonal Geometries:

Canonical Forms,

The Orthogonal Group,

The Witt Theorems for Orthogonal Geometries,

Maximal Hyperbolic Subspaces of an Orthogonal Geometry

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Metric Spaces :-

The Definition

Open and Closed Sets,

Convergence in a Metric Space,

The Closure of a Set,

Dense Subsets,

Continuity,

Completeness,

Isometries,

The Completion of a Metric Space.

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Hilbert Spaces :-

A Brief Review,

Hilbert Spaces,

Infinite Series,

An Approximation Problem,

Hilbert Bases,

Fourier Expansions,

A Characterization of Hilbert Bases,

Hilbert Dimension,

A Characterization of Hilbert Spaces,

The Riesz Representation Theorem.

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Tensor Products :-

Universality,

Bilinear Maps,

Tensor Products,

When Is a Tensor Product Zero?,

Coordinate Matrices and Rank,

Characterizing Vectors in a Tensor Product,

Defining Linear Transformations on a Tensor Product,

The Tensor Product of Linear Transformations,

Change of Base Field,

Multilinear Maps and Iterated Tensor Products,

Tensor Spaces,

Special Multilinear Maps,

Graded Algebras,

The Symmetric and Antisymmetric

The Determinant.

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Positive Solutions to Linear Systems :-

Convexity and Separation,

Convex, Closed and Compact Sets,

Convex Hulls,

Linear and Affine Hyperplanes,

Separation.

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Affine Geometry :-

Affine Combinations,

Affine Hulls,

The Lattice of Flats,

Affine Independence,

Affine Transformations,

Projective Geometry.

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