Linear Algebra Part 2

Basis And Dimension

What you’ll learn
Vector Spaces specifies the no. of independent directions in the space. Infinite dimensional Vector Space are Function spaces in Mathematical Analysis.
Students can enhance their knowledge in the study of Norms, Inner product Spaces, and other fundamental in mathematical analysis such as Banach & Hilbert spaces
To understand how students make sense of subspaces of vector spaces in series of in-depth qualitative interviews in technology assisted learning Environment.
Vector Spaces may be generalized in several ways, leading to more advanced notations in Abstract Algebra also offer a framework for Fourier Expansion.

Requirements
Basic Linear Algebra (Vector Spaces)

Description
LINEAR ALGEBRA- Part 2 (VECTOR SPACES)BASIS AND DIMENSIONWelcome to this 9+ hours of course on Linear Algebra where you will learn the Concept of Vector Spaces , Subspaces of Vector Spaces , Generators of Vectors , Linear Span , Linearly Dependent and Linearly Independent Vectors and Functions. In Linearly dependent and independent vectors and functions you will learn the concept of how to check whether vectors are linearly dependent or not. Furthermore you will learn the conditions of Trivial and Non Trivial Solutions along with Direct Sum of Subspaces. Then comes the Introduction to Basis and Dimension. In this section, some of the basic concepts in the study of vectors spaces is introduced. Basis is a linearly independent spanning set. The content on Basis for matrices, symmetric matrices, Hermitian Matices, Real Matrices, Columns of Invertible matrices and many more concepts. This course also provides the knowledge about how to find the Dimensions of Vector Spaces and Subspaces. The concept of Coordinate Vector is also included.This course is also subjected with so many Assignments covering the Definitions, Remarks, Notes , Postulates, with all the Expected Examples and Expected Theorems with Corollary. The assignments will helps you to get grasp of the subject.Vector Spaces are the subject of Linear Algebra and are well characterized by their dimension , which specifies the number of independent directions in the space. Infinte-Dimensional vector spaces arise naturally in Mathematical Analysis as function spaces, whose vectors are functions.For any queries related to the course, I would be happy to assist you. Just ping me via Inbox. You will get a course completion certificate after finishing the course.

Overview
Section 1: Introduction to Basis and Dimensions

Lecture 1 Assignment 1 Linearly Dependent and Linearly Independent Functions

Lecture 2 Assignment 2 Solved Examples

Lecture 3 Assignment 3 Examples to solve for Linearly Dependent or Independent functions

Section 2: Basis

Lecture 4 Assignment 1

Lecture 5 Assignment 2

Lecture 6 Assignment 3

Lecture 7 Assignment 4

Lecture 8 Assignment 5

Section 3: Basis for Matrices

Lecture 9 Basis for Matrices

Lecture 10 Basis for Hermitian Matrices

Lecture 11 Basis for all 2 by 3 Real Matrices

Lecture 12 Expected Example on Matrices

Lecture 13 Solved Examples on Matrices

Lecture 14 {1,1+x,1+x²} forms Basis for vector Space

Lecture 15 {1,1+x,1+x²,1+x³,………..} forms Basis for Vector Space

Lecture 16 {1,2-x,3+x²,4-x³} forms Basis for Vector Space

Lecture 17 Columns of Invertible n by n Matrix forms Basis for F(n by 1), Column Matrix.

Section 4: Basis For Vector Space

Lecture 18 Assignment 1

Lecture 19 Assignment 2

Lecture 20 Assignment 3

Lecture 21 Assignment 4

Lecture 22 Assignment 5

Section 5: Dimension of Vector Space V

Lecture 23 Definition of Dimension of a Vector Space

Lecture 24 No subset of V which contains less than n vectors can span V.

Lecture 25 Generating Set with n elements is a Basis for V

Lecture 26 Existence Theorem: A Linearly Independent Set with n elements is a Basis for V

Lecture 27 If W is a Subspace of Finite Dimensional V then dimW is than equal to dimV

Lecture 28 If W is a Proper Subspace of V, then dimW less than dimV

Lecture 29 If W1 and W2 be the Subspaces of V, then dim(W1+W2) =dimW1+dimW2-dim(intersec.)

Lecture 30 If intersection of W1&W2 is {0], then dim(W1+W2) = dimW1+dimW2

Section 6: Ordered Basis

Lecture 31 Assignment 1 Solved Examples

Lecture 32 Assignment 2 Solved Examples

Lecture 33 Assignmnet 3 Direct Sum of Subspaces(1)

Lecture 34 Assignment 4 Direct Sum of Subspaces(2)

Lecture 35 Assignment 5 Definition of Coordinate Vector with Solved Examples

Students of mathematics , Physics and Engineering, Graduates and Post Graduates Students.