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Linear Algebra Part 3 : VECTOR SPACES & RANK of MATRICES

Basis And Dimension, vector spaces, row space , row rank of matrix, row echelon matrix, Linear Algebra, column space

Created byJaswinder Kaur

Last updated 2/2025

English

What you'll learn

How to get Row Echelon matrix , Column echelon Matrix, Elementray matrices, Row Space and Row Rank of matrix, Column Space and Column rank of Matrix.
How to find the Basis and Dimension of Row Space or Column Space of Matrix
How to find the Basis of Vector Space generated by given Polynomials and also by given Matrices.
Coordinate Vector of a vector relative to the Standard Basis.

Course content

5 sections • 45 lectures • 5h 22m total length

Introduction to Leading Entry & Column Index14:02
Introduction to Row Echelon Matrix12:30
Elementary Row Matrices11:19
Theorem on Elementary Column Matrices2:34
Row Space & Row Rank of a Matrix7:41
Show that R(PA) = R(A) if P is a Non Singular Matrix7:28
Show that Row Rank(PA) = Row Rank (A) If P is a Non Singular Matrix8:00
Prove that R(A) = R(B) & consequently Row Rank(A) = Row Rank(B).7:18

Number of non zero rows of an Row Echelon Matrix equals to the Row rank of A9:46
Basis for the Row Space of an Equivalent matrix6:18
Expected Theorem : Equivalent statements for two Matrices10:40
Matrix A is Non Singular if and only if Matrix B is Non singular5:08
Examples of matrices having equal Row Rank and vector common to their row space6:37
If R(A) subset of R(B) then R(A) = R(B) if A is non singular.4:14

Introduction to Column Space and Column rank of a Matrix6:50
Show that C(AQ) is a subset of C(A) & C(AQ) = C(A) if a is Non singular Matrix.5:52
Show that Column Rank(AQ) = Column Rank(A) if A is Non singular Matrix6:01
Show that C(A) = C(B) & Column Rank (A) = Column Rank (B)5:27
Column Rank of a Column Echelon Matrix D is equal to its no. of non zero rows.7:51
Non zero Columns of D which is Column equivalent to A forms Basis for C(A)7:46
Equivalent Statements for two Matrices8:00
If C(A) = C(B) then A is Non singular if and only if B is Non Singular4:40
If C(A) is subset of C(B) then show that C(A) = C(B) if A is Non singular Matrix5:22
Give an example of A & B matrices s.t. C(A) = C(B) & R(A) = R(B)4:03

Find the Basis for Row Space of Matrix9:52
Find the Basis and Dimension of Row Space of Matrix.10:21
Find the Basis and Dimension of Row Space of Matrix where, A is 3 by 3 Matrix.4:57
Prove that R(A) = R(B) for two given matrices A & B.9:07
Find the Basis for the Row Space of Matrix generated by the given Vectors.6:49
Prove that two Vector Spaces generated by different Vectors are Equal.10:09
Show that space V is equal to the Space W generated by different Vectors.10:08
Assignment Based on Basis & Dimension provided with Answers.2:14
In R³, find the basis for xz-plane.4:06
In R³, find the Basis and Dimension for xy-plane.2:46
In R³, find the Basis and Dimension yz-plane.3:33
Find the Basis and Dimension for z-axis.3:04
Find the Basis and Dimension for (a,0,b) for all reals 'a' & 'b'.2:57
Find the Basis & Dim for W having the vectors whose first component is 0.4:19

Introduction to the Coordinate Vector relative to the Standard Ordered Basis5:31
Eg 1)Find the Basis of the Vector Space generated by the Given Polynomials9:09
Eg 2)Find the Basis of the Vector Space generated by the Given Polynomials10:23
Show that the Vector Spaces V & W generated by given Polynomials are Equal.15:28
Eg3) Find the Basis for Subspace W of V generated by given Matrices.10:17
Eg4) Find the Basis for Subspace W of V generated by given Matrices9:01
Assignment on Basis & dimension on Coordinate Vectors.2:54

Requirements

Basic Knowledge of Vector Spaces and Subspaces.

Description

Introducing this 5 hour 23 minute Course on ROW SPACE AND ROW RANK OF MATRIX

The Course includes_

The concepts of leading entries of rows and column indices.

The definition for Row Echelon Matrix and its properties.

The examples of Row Echelon Matrices and definition of Elementary Row Operations and Elementary Column Operations.

Methods of reducing a matrix into row echelon form

To get Column echelon form matrix from given matrix.

Definition of Row Space and Row rank of matrix .

Definition of Column Space and Column Rank of matrix.

Theorems on Row space and row rank of matrix.

Theorems on Column space and column rank of matrix.

The equivalent statements on matrices for row space and column space of matrix

To get Basis and Dimension of row space of matrix and

To get Basis and Dimension of column space of matrix.

Basis for the vector space generated by Polynomials.

Basis for the vector space generated by matrices.

Concept of Coordinate Vectors relative to standard basis and Solved Examples.

And, To get Basis for vector space generated by given polynomials or matrices using Coordinate vectors .

Assignments on Basis and Dimensions including all the Expected theorems and all types of examples based on the above contents.

Thank You

Who this course is for:

Bachelor of science students, Master of mathematical sciences, gradiuate and post graduate students, students of linear algebra, students of vector spaces linear algebra, linear algebra

Linear Algebra Part 3 : VECTOR SPACES & RANK of MATRICES

What you'll learn

Explore related topics

Course content

Row Echelon matrix , Row space & Row Rank of a Matrix8 lectures • 1hr 11min

Theorems on Row Echelon Matrix and Row rank of Matrix6 lectures • 43min

Column Space & Column Rank of a Matrix (Unit 2)10 lectures • 1hr 2min

Basis and Dimension of Matrix for Row Space & Column Space14 lectures • 1hr 24min

Coordinate Vectors Relative to Basis7 lectures • 1hr 3min

Requirements

Description

Who this course is for: