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.

3

3

-

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.

3

3

-

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.

3

3

-

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.

3

3

-

Eigenvalues and Eigenvectors :-

Geometric and Algebraic Multiplicities.

The Jordan Canonical Form.

Triangularizability and Schur's Theorem.

Diagonalizable Operators.

3

3

-

Real and Complex Inner Product Spaces,Norm and Distance,Isometries,

Orthogonality,Orthogonal and Orthonormal Sets.

The Projection Theorem and Best Approximations.

The Riesz RepresentationTheorem.

3

3

-

Structure Theory for Normal Operators :-

The Adjoint of a Linear Operator,Orthogonal ProjectionsUnitary 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.

3

3

-

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.

3

3

-

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.

3

3

-

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.

3

3

-

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.

3

3

-

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.

3

3

-

Positive Solutions to Linear Systems :-

Convexity and Separation,Convex, Closed and Compact Sets,Convex Hulls,

Linear and Affine Hyperplanes,Separation.

3

3

-

Affine Geometry :-

Affine Combinations,Affine Hulls.

The Lattice of Flats.

Affine Independence,Affine Transformations.

Projective Geometry.

3

3

-