Linear Algebra Part 4 (Echelon Matrix & Normal Form Matrix)



Echelon matrix , Normal Form of matrix, linear algebra, vector spaces, basis and dimension , Rank of matrix

What you'll learn
Knowledge of Echelon Matrices and Normal form of Matrix
Determining the Basis and Dimension of Subspaces, Sum of Subspaces and Intersection of Subspaces including the Rank
Elementary Row and Column Operations on Matrices
Determining the Non Singular Matrices by reducing the Matrix into Normal Form.

Requirements
Basic knowledge of Matrices

Description
Linear Algebra, mathematical discipline that deals with vectors and matrices and, more generally, with vector spaces and linear transformations. In this 3hr 54 min Course ' Linear Algebra Part 4 Echelon Matrix and Normal Form of Matrix' is having very interesting contents based on Echelon Matrix, Row Column Operations on matrix, Rank of Matrix, Normal Form of Matrix, and Determining the Non singular Matrices.The listed Contents of the Course 'Echelon Matrix & Normal Form of Matrix'1) The introduction to the Echelon Matrix and its definition with examples.2) Finding the Basis and Dimension of subspaces.3) Finding basis and dimension of the sum of subspaces.4) Finding the basis and dimension of intersection of subspaces.5) Finding the basis and dimension of subspaces having vectors as matrices.6) Finding the basis and dimension of subspaces having vectors as real polynomials of degree less than equal to 3 including the zero polynomial.7) Finding the basis and dimension of subspaces, having vectors as xy-plane or x axis or respective other axis and planes. Finding the basis and dimension of subspaces, sum of subspaces, intersection of subspaces with determination of rank too.9) Equivalence of row column operations on matrices.10) Normal form of matrix introduction with examples11) Determining the rank of matrix by reducing the given matrix into its normal form.12)Determining the non singular matrices P and Q by reducing the given matrix into its normal form such that PAQ is in normal form where A is the given matrix.Including all Important Theorems and Proofs with Solved Examples and assignments plus Practice Questions.

Overview
Section 1: Echelon Matrix

Lecture 1 Introduction to Echelon Matrix with Examples

Lecture 2 Find the Basis & Dimension of Subspaces & Sum of Subspaces

Lecture 3 Find the Basis & Dimension of Subspaces, Sum of Subspaces & Itheir Intersection

Lecture 4 Find the Basis & Dimension of the subspaces for including vectors (a,0,b)

Lecture 5 Find the Basis & Dimension of Subspaces for xz-plane.

Lecture 6 Find the Basis & Dimension of xy-plane and y axis

Lecture 7 Show that Sum of the Subspaces is R³

Section 2: Extension of Basis

Lecture 8 Find the Basis & Dimension of Subspace and extend its Bssis

Lecture 9 Extend the set { (1,1,1,1),(1,2,1,2) } to form Basis for R4

Lecture 10 Extend the set {(0,0,1,2,3),(0,0,-2,1,2)} to form Basis for R5

Lecture 11 Find the Dimension of Sum and Intersection of Subspaces for Real Polynomials

Lecture 12 Find the Basis & Dimension of W = {f(x)/ f(1) = 0} and also extend its Basis

Lecture 13 Find the Basis & Dimension of W = {f(x)/ f'(1) = 0} and also extend its Basis

Lecture 14 Find the Basis & Dimension of Sum of Subspaces for previous content Polynomials

Lecture 15 Find Basis & Dim of Intersection of Subspaces for previous content Polynomials

Section 3: Basis and Dimension of Solution Space

Lecture 16 Basis and Dimension of Solution Space of system of Linear Equations

Lecture 0 Basis and Dimension of Solution Space of system of 3 Linear Equations

Lecture 0 Basis & Dimension of W = {(x,y,z)/x-2y+3z = 0} and also extend its Basis.

Lecture 0 Basis & Dimension of subspace having vectors (x,y,z,s) s.t. y = x-z, x = 2s

Lecture 17 Basis & Dimension of Subspace W = {[x,y,z)/z = x+y, y = 2x}

Lecture 18 Basis & dimension of Sum and Intersection of given Subspaces of R4

Lecture 19 Basis & dimension of Sum and Intersection of given Subspaces of R4 (Exercise 2)

Lecture 20 Basis & dimension of Sum and Intersection of given Subspaces of R4 (Exercise 3)

Lecture 21 Determine whether given polynomials are Linearly Independent or Dependent.

Lecture 22 Determine whether given Matrices are Linearly Independent or Dependent.

Section 4: Rank of a Matrix

Lecture 23 Introduction to Rank of Matrix

Lecture 24 Theorem 1 on Rank of Matrix

Lecture 25 Theorem 2 on Rank of Matrix ( Equivalent Statements)

Lecture 26 Show that row column operation is an Equivalence Relation

Lecture 27 Important Results on rank of Matrix

Section 5: Normal Form of Matrix

Lecture 28 Introduction to Normal Form of Matrix_ Reduction to Normal Form of Matrix

Lecture 29 Reduction to Normal Form of Matrix illustrating with an Example.

Lecture 30 Reduce the given Matrix into its Normal Form and also Determine its Rank

Lecture 31 Prove that Row Rank of A = Rank of A = Column Rank of A where A is given Matrix

Lecture 32 Find the Rank of Matrix by reducing this matrix into its Normal Form

Lecture 33 Find the Non Singular Matrices P & Q s.t. PAQ is in Normal Form

Lecture 34 Practice Assignment to find the Non Singular Matrices

Bsc. and Msc Maths students, for UGC NET EXAM Entrance Exam, for CSIR NET Exam, Engineering Higher Mathematics students, Post Graduate students

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