Course

Unit 1

Fields, System of Linear Equations, Matrices and Elementary Row Operations, Row Reduced Echelon Matrices, Elementary Matrices, Invertible Matrices. Vector Space over a Field, Definition and Examples, Subspaces, Spanning set, Linear Independence, Basis and Dimension of a Vector Space.

Unit 2

Ordered Basis and Coordinates, Row Space and Row Equivalent Matrices. Linear Transformations: Properties, Rank and Nullity of a Linear transformation, Algebra of Linear Transformations, Isomorphism of Vector Spaces, Representation of Linear Transformations by Matrices, Similar Matrices.

Unit 3

Linear Functionals, Dual Space, Annihilators of subspaces, Transpose of a Linear Transformation, Characteristics value and Characteristic polynomial of a Linear Operator, Minimal and Characteristic Polynomial.

Unit 4

Cayley Hamilton Theorem, Invariant Subspaces of an Operator, Diagonalizability of an Operator, Simultaneous Diagonalization.

Unit 5

Direct Sum Decompositions, Invariant Direct Sums, Primary Decomposition Theorem, Cyclic Subspaces and Annihilators, Cyclic Decomposition Theorem and Rational Form, Jordan Form.

CO1: Ability to understand the basic concepts of vector and matrix algebra, including linear dependence / independence, basis and dimension of a subspace, for analysis of matrices and systems of linear equations

CO2: Ability to find the dimension of spaces such as those associated with matrices and linear transformations

CO3: Ability to understand Dual Space, subspaces, sub space of a linear transformation Minimal and Characteristic Polynomial

CO4: To understand the construction of matrices for a linear transformation in the triangular/ Jordan form

CO5: Apply the decomposition theorem in context of mathematical applications to subspaces

Textbook
K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall.

References

• Sheldon Axler, Linear Algebra Done Right, 2nd Edition, Springer.
• Dummit and Foote, Abstract Algebra, Wiley.