Linear Algebra Part 4 (Echelon Matrix & Normal Form 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.8) 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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