Linear Algebra Part 2 : VECTOR SPACES ( BASIS & DIMENSION)

Name: Linear Algebra Part 2 : VECTOR SPACES ( BASIS & DIMENSION)
Rating: 4.5 (1 reviews)

Basis And Dimension, Linear Algebra, Vector spaces, dimension of subspace, Types of matrices, Coordinate 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

6 sections • 35 lectures • 7h 15m total length

Assignment 1 Linearly Dependent and Linearly Independent Functions27:21
Assignment 2 Solved Examples13:39
Assignment 3 Examples to solve for Linearly Dependent or Independent functions9:37

Basis for Matrices14:16
Basis for Hermitian Matrices9:51
Basis for all 2 by 3 Real Matrices7:09
Expected Example on Matrices12:42
Solved Examples on Matrices9:19
{1,1+x,1+x²} forms Basis for vector Space8:51
{1,1+x,1+x²,1+x³,...........} forms Basis for Vector Space8:23
Prove that the polynomials {1, 1+x, 1+x+x^2, ..., 1+x+...+x^n} form a basis for the real polynomials of degree at most n by establishing linear independence and spanning.
{1,2-x,3+x²,4-x³} forms Basis for Vector Space7:33
Demonstrate that the polynomials 1, 2−x, 3+x^2, 4−x^3 form a basis for real polynomials degree ≤3 by proving linear independence and generating all vectors in V.
Columns of Invertible n by n Matrix forms Basis for F(n by 1), Column Matrix.9:19
Demonstrate that the columns of an invertible n by n matrix form a basis for the space of f n by 1 column matrices, by proving linear independence and spanning.

Definition of Dimension of a Vector Space13:52
Defines the dimension of a finite vector space as the number of vectors in any basis, with examples like R3 is 3-dimensional and polynomials degree ≤ n have dimension n+1.
No subset of V which contains less than n vectors can span V.9:48
Demonstrate that in a finite dimensional vector space V with dim V = n, no spanning set contains fewer than n vectors. Use a basis to derive a contradiction.
Generating Set with n elements is a Basis for V11:57
Existence Theorem: A Linearly Independent Set with n elements is a Basis for V17:31
If W is a Subspace of Finite Dimensional V then dimW is than equal to dimV6:19
If W is a Proper Subspace of V, then dimW less than dimV8:23
If W1 and W2 be the Subspaces of V, then dim(W1+W2) =dimW1+dimW2-dim(intersec.)21:23
If intersection of W1&W2 is {0], then dim(W1+W2) = dimW1+dimW213:09

Assignment 1 Solved Examples13:12
Demonstrate basis construction for polynomials degree ≤ n using 1+x, 1+x^2, ..., 1+x^n, showing linear independence and dimension n+1; apply subspace intersection and sum dimensions to R3 examples.
Assignment 2 Solved Examples13:59
Assignmnet 3 Direct Sum of Subspaces(1)14:25
Extend w1's basis to a full basis of V to form w2, show V = w1 + w2 and w1 ∩ w2 = {0}, proving the direct sum of subspaces.
Assignment 4 Direct Sum of Subspaces(2)12:13
Assignment 5 Definition of Coordinate Vector with Solved Examples12:49

Requirements

Basic Linear Algebra (Vector Spaces)

Description

LINEAR ALGEBRA- Part 2 (VECTOR SPACES)

BASIS AND DIMENSION

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 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 Matrices, 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.

Who this course is for:

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

Linear Algebra Part 2 : VECTOR SPACES ( BASIS & DIMENSION)

What you'll learn

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Course content

Introduction to Basis and Dimensions3 lectures • 51min

Basis5 lectures • 1hr 20min

Basis for Matrices9 lectures • 1hr 27min

Basis For Vector Space5 lectures • 49min

Dimension of Vector Space V8 lectures • 1hr 42min

Ordered Basis5 lectures • 1hr 7min

Requirements

Description

Who this course is for: