Linear Algebra Part 3 (VECTOR SPACES AND SUBSPACES)

Course Curriculum

Chapter 1: Row Echelon matrix , Row space & Row Rank of a Matrix

Lecture 1: Introduction to Leading Entry & Column Index

Lecture 2: Introduction to Row Echelon Matrix

Lecture 3: Elementary Row Matrices

Lecture 4: Theorem on Elementary Column Matrices

Lecture 5: Row Space & Row Rank of a Matrix

Lecture 6: Show that R(PA) = R(A) if P is a Non Singular Matrix

Lecture 7: Show that Row Rank(PA) = Row Rank (A) If P is a Non Singular Matrix

Lecture 8: Prove that R(A) = R(B) & consequently Row Rank(A) = Row Rank(B).

Chapter 2: Theorems on Row Echelon Matrix and Row rank of Matrix

Lecture 1: Number of non zero rows of an Row Echelon Matrix equals to the Row rank of A

Lecture 2: Basis for the Row Space of an Equivalent matrix

Lecture 3: Expected Theorem : Equivalent statements for two Matrices

Lecture 4: Matrix A is Non Singular if and only if Matrix B is Non singular

Lecture 5: Examples of matrices having equal Row Rank and vector common to their row space

Lecture 6: If R(A) subset of R(B) then R(A) = R(B) if A is non singular.

Chapter 3: Column Space & Column Rank of a Matrix (Unit 2)

Lecture 1: Introduction to Column Space and Column rank of a Matrix

Lecture 2: Show that C(AQ) is a subset of C(A) & C(AQ) = C(A) if a is Non singular Matrix.

Lecture 3: Show that Column Rank(AQ) = Column Rank(A) if A is Non singular Matrix

Lecture 4: Show that C(A) = C(B) & Column Rank (A) = Column Rank (B)

Lecture 5: Column Rank of a Column Echelon Matrix D is equal to its no. of non zero rows.

Lecture 6: Non zero Columns of D which is Column equivalent to A forms Basis for C(A)

Lecture 7: Equivalent Statements for two Matrices

Lecture 8: If C(A) = C(B) then A is Non singular if and only if B is Non Singular

Lecture 9: If C(A) is subset of C(B) then show that C(A) = C(B) if A is Non singular Matrix

Lecture 10: Give an example of A & B matrices s.t. C(A) = C(B) & R(A) = R(B)

Chapter 4: Basis and Dimension of Matrix for Row Space & Column Space

Lecture 1: Find the Basis for Row Space of Matrix

Lecture 2: Find the Basis and Dimension of Row Space of Matrix.

Lecture 3: Find the Basis and Dimension of Row Space of Matrix where, A is 3 by 3 Matrix.

Lecture 4: Prove that R(A) = R(B) for two given matrices A & B.

Lecture 5: Find the Basis for the Row Space of Matrix generated by the given Vectors.

Lecture 6: Prove that two Vector Spaces generated by different Vectors are Equal.

Lecture 7: Show that space V is equal to the Space W generated by different Vectors.

Lecture 8: Assignment Based on Basis & Dimension provided with Answers.

Lecture 9: In R³, find the basis for xz-plane.

Lecture 10: In R³, find the Basis and Dimension for xy-plane.

Lecture 11: In R³, find the Basis and Dimension yz-plane.

Lecture 12: Find the Basis and Dimension for z-axis.

Lecture 13: Find the Basis and Dimension for (a,0,b) for all reals 'a' & 'b'.

Lecture 14: Find the Basis & Dim for W having the vectors whose first component is 0.

Chapter 5: Coordinate Vectors Relative to Basis

Lecture 1: Introduction to the Coordinate Vector relative to the Standard Ordered Basis

Lecture 2: Eg 1)Find the Basis of the Vector Space generated by the Given Polynomials

Lecture 3: Eg 2)Find the Basis of the Vector Space generated by the Given Polynomials

Lecture 4: Show that the Vector Spaces V & W generated by given Polynomials are Equal.

Lecture 5: Eg3) Find the Basis for Subspace W of V generated by given Matrices.

Lecture 6: Eg4) Find the Basis for Subspace W of V generated by given Matrices

Lecture 7: Assignment on Basis & dimension on Coordinate Vectors.