Bsc 6th sem m-1 full notes
Course Title: 6.1Linear Algebra
Sem. VI
Formative Summative Duration of
Assessment Assessment ESA:
Marks: 40 Marks: 60 02 hrs.
Course Course Learning Outcomes: The overall expectation from this course is that the Outcomes student will build a basic understanding in few areas of linear algebra such as vectors spaces, linear transformations. Some broader course outcomes are listed as follows. At the end of this course, the student will be able to
- Understand the concepts of Vector spaces, subspaces, bases dimension and their
properties.
- Become familiar with the concepts of Eigen values and Eigen vectors, linear
transformations etc.
- Prove various statements in the context of vectors spaces.
Unit I
Rings and integral domains: Rings, Properties of rings, sub rings,
ideals, principal and maximal ideals in commutative ring, quotient
ring, homomorphism and isomorphism, and integral domains.
Unit II
Vector spaces: Definition, examples and properties; Subspaces -
Examples, criterion for a sub-set to be a subspace and some properties;
Linear Combination-Linear span, Linear dependence and Linear
independence, basic properties of linear dependence and
independence, techniques of determining linear dependence and
independence in various vector spaces and related problems; Basis and
dimension - Co-ordinates, ordered basis, some basic properties of
basis and dimension and subspace Spanned by given set of vectors;
Quotient space. Dimension of quotient space (derivation in finite
case); Sum and Direct sum of subspaces - Dimensions of sum and
direct sum spaces (Derivation in finite case).
Unit III
Linear transformations: Definition, examples, equivalent criteria,
some basic properties and matrix representation and change of basis
and effect on associated matrix, similar matrices; Rank - Nullity
theorem - Null space, Range space, proof of rank nullity theorem and
related problems.
Unit IV
Isomorphism, Eigen values and Diagonalization:
Homomorphism, Isomorphism and automorphism-Examples, order of
automorphism and Fundamental theorem of homomorphism; Eigen
values and Eigen vectors-Computation of Eigen values, algebraic
multiplicity, some basic properties of Eigen values, determination of
eigenvectors and Eigen space and geometric multiplicity.
Diagonalizability of linear transformation - Meaning, condition based
on algebraic and geometric multiplicity (mentioning) and related
problems(Only verification of diagonalizability).
Recommended Leaning Resources
References:
1. I. N. Harstein, Topics in Algebra, 2nd Edition,Wiley.
2. Stephen H. Friedberg, Arnold J. Insel & Lawrence E. Spence (2003), Linear
Algebra (4th Edition), Printice- Hall of India Pvt. Ltd.
3. F. M. Stewart, Introduction to Linear Algebra, Dover Publications.
4. S .Kumaresan, LinearAlgebra, Prentice Hall India Learning Private Limited.
5. Kenneth Hoffman & Ray Kunze (2015), Linear Algebra, 2ndEdition), Prentice Hall India Leaning Private Limited.
6. Gilbert. Strang (2015), Linear Algebra and its applications, (2ndEdition), Elsevier.
7. Vivek Sahai & Vikas Bist(2013), Linear Algebra (2nd Edition) Narosa Publishing.
8. Serge Lang (2005), Introduction to Linear Algebra (2ndEdition), Springer India.
9. T. K. Manicavasagam Pillaiand K S Narayanan, Modern Algebra Volume2.
Practical/Lab Work to be performed in Computer Lab (FOSS)
Suggested Software’s: Maxima/ Scilab /Python/R.
Suggested Programs:
1. Program on multiple product of vectors–Scalar and Cross product.
2. Program on vector differentiation and finding unit tangent.
3. Program to find curvature and torsion of a space curve.
4. Program to find the gradient and Laplacian of a scalar function,
divergence and curl of a vector function.
5. Program to demonstrate the physical interpretation of gradient, divergence
and curl.
6. Program to evaluate vector line integral.
7. Program to evaluate a surface integral.
8. Program to evaluate a volume integral.
9. Program to verify Green’s theorem.
10. Program to find equation and plot sphere, cone and cylinder
11. Program to find distance between a straight line and a plane.
12. Program to construct and plot some standard surfaces.
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