Linear Algebra Part 3 (VECTOR SPACES AND SUBSPACES)

سرفصل های دوره

Basis And Dimension, vector spaces, row space and row rank of matrix, row echelon matrix, coordinate vectors,,

1. Row Echelon matrix , Row space & Row Rank of a Matrix

1. Introduction to Leading Entry & Column Index

2. Introduction to Row Echelon Matrix

3. Elementary Row Matrices

4. Theorem on Elementary Column Matrices

5. Row Space & Row Rank of a Matrix

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

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

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

2. Theorems on Row Echelon Matrix and Row rank of Matrix

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

2. Basis for the Row Space of an Equivalent matrix

3. Expected Theorem Equivalent statements for two Matrices

4. Matrix A is Non Singular if and only if Matrix B is Non singular

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

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

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

1. Introduction to Column Space and Column rank of a Matrix

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

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

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

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

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

7. Equivalent Statements for two Matrices

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

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

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

4. Basis and Dimension of Matrix for Row Space & Column Space

1. Find the Basis for Row Space of Matrix

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

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

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

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

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

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

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

9. In R, find the basis for xz-plane.

10. In R, find the Basis and Dimension for xy-plane.

11. In R, find the Basis and Dimension yz-plane.

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

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

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

5. Coordinate Vectors Relative to Basis

1. Introduction to the Coordinate Vector relative to the Standard Ordered Basis

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

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

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

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

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

7. Assignment on Basis & dimension on Coordinate Vectors.

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