FOURTH SEMESTER MATHEMATICS
LINEAR ALGEBRA
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 subspace, Algebra of subspace, Linear Sum of two subspace, 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 a 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, Orthogonal set, complete orthonormal set, Gram– Schmidt orthogonalisation process. Bessel’s inequality and Parseval’s Identity.