4th Semester Degree MATHEMATICS LINEAR ALGEBRA Material

Course Outcomes:

After successful completion of this course, the student will be able to;

1. Understand the concepts of vector spaces, subspaces, basises, dimension and their properties

2. Understand the concepts of linear transformations and their properties

3. Apply Cayley- Hamilton theorem to problems for finding the inverse of a matrix and higher powers of matrices without using routine methods

4. learn the properties of inner product spaces and determine orthogonality in inner product spaces.

Course Syllabus:


UNIT – I Vector Spaces-I

Vector Spaces, General properties of vector spaces, n-dimensional Vectors, addition and scalar multiplication of Vectors, internal and external composition, Null space, Vector subspaces, Algebra of subspaces, Linear Sum of two subspaces, linear combination of Vectors, Linear span Linear independence and Linear dependence of Vectors.

UNIT –II Vector Spaces-II

Basis of Vector space, Finite dimensional Vector spaces, basis extension, co-ordinates, Dimension of vector space, Dimension of a subspace, Quotient space and Dimension of Quotient space.

UNIT –III Linear Transformations

Linear transformations, linear operators, Properties of L.T, sum and product of LTs, Algebra of Linear Operators, Range and null space of linear transformation, Rank and Nullity of linear transformations – Rank – Nullity Theorem.

UNIT –IV Matrix 

Matrices, Elementary Properties of Matrices, Inverse Matrices, Rank of Matrix, Linear Equations, Characteristic equations, Characteristic Values & Vectors of square matrix, Cayley – Hamilton Theorem.

UNIT –V Inner product space

Inner product spaces, Euclidean and unitary spaces, Norm or length of a Vector, Schwartz inequality, Triangle Inequality, Parallelogram law, Orthogonality, Orthonormal set, complete orthonormal set, Gram– Schmidt orthogonalisation process. Bessel’s inequality and Parseval’s Identity.

fourth semester MATHEMATICS LINEAR ALGEBRA Material

Name of the Unit Download Link
UNIT – I Vector Spaces-I Click Here
UNIT –II Vector Spaces-II Click Here
UNIT –III Linear Transformations Click Here
UNIT –IV Matrix Click Here
UNIT –V Inner product space Click Here