Linear Algebra- Part 1 (Vector Spaces and Subspaces)

Name: Linear Algebra- Part 1 (Vector Spaces and Subspaces)
Rating: 4.5 (4 reviews)

Vector Spaces, Basis and Dimension, Subspaces, Generators of vectors linear dependent & linearly independent vectors

Created byJaswinder Kaur

Last updated 2/2025

English

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.

Course content

5 sections • 31 lectures • 9h 28m total length

Vector Spaces Introduction18:18
Vector Spaces Introduction Continued.....25:10
Introduction of Subspaces20:51
Conditions to prove the Subset of vector space to be a Subspace .16:48
Sum of Two Subspaces is a Subspace13:20
Comparable Subspaces19:11
Set of All Hermition and Skew Hermition Matrices is not a Subspace28:00

Set of Even Functions and Odd Function is a subspace.26:42
Learn that the real-valued function space decomposes into two subspaces: even functions and odd functions, whose intersection is the zero function and whose sum equals the whole space.
Vector Spaces of all Real and Complex Valued Functions22:01
Solved Examples on Vector Spaces23:55
Solved Examples on Vector Spaces with Matrices14:12
Condition of Uniqueness of vectors from Subspaces13:57

Definition of Linear Combination of Vectors17:52
Linear Span and Generators of Vector Space22:03
Explore the linear span of a vector set and how it generates the smallest subspace of a vector space, using linear combinations to characterize generating sets and subspace containment.
Linear Span L(S) is the intersection of all Subspaces of V containing S13:53
Result : If S C T , then L(S) C L(T) with Expected Theorems20:16
Demonstrate that if S is a subset of T, then L(S) is contained in L(T) and prove the reverse to establish equality, with examples generating R^3 from given vectors.
Finitely Generated vector Space24:11
Expected Examples using Invertible matrix , Singular and non singular Matrices18:27
Exercise on lineraly Dependent and Linearly Independent vectors20:12
Subset of set of Linearly Independent vectors is also linearly Independent.12:45
Some Expected Prepositions12:32
Prepositions on Linearly Dependent and Linearly Independent set of vectors.15:27
Exercises using the Concept of Coplanar Vectors and Collinear Vectors .13:48
Every non trivial subset of V is Linearly Dependent and Expected Prepositions .12:24
Identify when a nonzero vector is a linearly independent singleton, two vectors are dependent iff one is a scalar multiple, and discuss extending a generating set while preserving generation.
Expected Solved Examples.12:50

Requirements

Concept of Basic Linear Algebra .

Description

Linear Algebra- Part 1 (Vector Spaces)

Welcome 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. In Linearly dependent and independent vectors 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. 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.

Who this course is for:

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

Linear Algebra- Part 1 (Vector Spaces and Subspaces)

What you'll learn

Explore related topics

Course content

Introduction of Vector Spaces and Subspaces with Assignments7 lectures • 2hr 22min

Vector Spaces (Theorems and Assignment)5 lectures • 1hr 41min

Direct Sum5 lectures • 1hr 49min

Basis And Dimension(Introduction)13 lectures • 3hr 37min

Course Wrap up1 lecture • 1min

Requirements

Description

Who this course is for: