Recommended Algebra Books to read? 
Can somebody recommend me any books to read that cover the following
  topics?

Chapter 1. Vector Spaces


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*1.1. Solutions of Simultaneous Linear Equations

*1.2. Fields and Vector Spaces

*1.3. Product of Sets and of Vector Spaces

*1.4. Vector Subspaces

*1.5. Linear Independence and Bases

*1.6. Dimension of a Vector Space

*1.7. Linear Mappings

*1.8. Rank-Nullity Theorem


Chapter 2. Linear Mappings and Matrices


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*Linear Mappings $F^m → F^n$ and Matrices

*Basic Properties of Matrices 27

*Abstract Linear Mappings and Matrices 30

*Change of a Matrix by Change of Basis 33


Chapter 3. Rings and Modules 37
Chapter 4. Determinants and Eigenvalues Redux 63
Chapter 6. Jordan Normal Form


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*Statement of the Jordan Normal Form and Strategy of Proof

*The proof of Jordan Normal Form

*Example of Jordan Normal Form

*A Brief Explanation of the Final Step in PageRank as an Application of the Jordan


I have grown really fond of Axler's "Linear Algebra Done Right." It focuses more on conceptual problems and not tedious calculations, and emphasizes important concepts that come into play in other subjects later on. This covers most of what you mentioned in your post, for those that you don't find you can always read Manin's "Linear Algebra."
