Utkal University, Odisha, Mathematics Semester 5, Linear Algebra Syllabus

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Unit - 1 Vector Spaces

Unit 1

Vector spaces

1.1 Vector spaces subspaces and examples

1.2 Algebra of subs paces quotient spaces

1.3 Linear combination of vectors linear span linear independence

1.4 Basis and dimension dimension of subspaces

1.5 Linear transformations null space range

1.6 Rank and nullity of a linear transformation

Unit - 2 Matrix Representation Of A Linear Transformation

Unit 2

Matrix representation of a linear transformation

2.1 Matrix representation of a linear transformation

2.2 Algebra of linear transformations

2.3 Isomorphisms Isomorphism theorems invertibility and isomorphisms

2.4 Change of coordinate

2.5 Matrix Dual spaces dual basis double dual

2.6 Transpose of a linear transformation and its matrix in the dual basis

2.7 Annihilators Basics of Fields

Unit - 3 Eigenspaces Of A Linear Operator

Unit – 3

Eigenspaces of a linear operator

3.1 Eigenspaces of a linear operator diagonalizability

3.2 Invariant subspaces and CayleyHamilton theorem

3.3 The minimal polynomial for a linear operator

3.4 Inner product spaces and norms

3.5 Gram Schmidt orthogonalization process

Unit - 4 Orthogonal Complements

Unit 4

Orthogonal complements

4.1 Orthogonal complements Bessels inequality

4.2 The adjoint of a linear operator

4.3 Least Squares Approximation

4.4 Minimal solutions to systems of linear equations

4.5 Normal and selfadjoint operators

4.6 Orthogonal projections and Spectral theorem

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Other Subjects of Semester-5
Linear programming
Multivariate calculus
Probability and statistics
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