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1. Introduction to the Course Contents.

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10. Subspaces (Continued) , Spanning Sets, Linear Independence, Dependence

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11. Basis for a Vector Space

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12. Dimension of a Vector Space

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13. Dimensions of Sums of Subspaces

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14. Linear Transformations

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15. the Null Space and the Range Space of a Linear Transformation

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16. the Rank-Nullity-Dimension Theorem. Isomorphisms between Vector Spaces

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17. Isomorphic Vector Spaces, Equality of the Row-Rank and the Column-Rank I

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18. Equality of the Row-Rank and the Column-Rank II

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19. the Matrix of a Linear Transformation

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2. Linear Equations

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20. Matrix for the Composition and the Inverse. Similarity Transformation

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21. Linear Functionals. the Dual Space. Dual Basis I

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22. Dual Basis II. Subspace Annihilators I

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23. Subspace Annihilators II

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24. the Double Dual. the Double Annihilator

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25. the Transpose of a Linear Transformation. Matrices of a Linear Transformation and its Transpose

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26. Eigenvalues and Eigenvectors of Linear Operators

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27. Diagonalization of Linear Operators. a Characterization

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28. the Minimal Polynomial

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29. the Cayley-Hamilton Theorem

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2Nd Uniqueness Theorem, Artinian Rings

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30. Invariant Subspaces

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31. Triangulability, Diagonalization in Terms of the Minimal Polynomial

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32. Independent Subspaces and Projection Operators

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33. Direct Sum Decompositions and Projection Operators I

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34. Direct Sum Decomposition and Projection Operators II

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35. the Primary Decomposition Theorem and Jordan Decomposition

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36. Cyclic Subspaces and Annihilators

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37. the Cyclic Decomposition Theorem I

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38. the Cyclic Decomposition Theorem II. the Rational Form

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39. Inner Product Spaces

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3a. Equivalent Systems of Linear Equations I: Inverses of Elementary Row-Operations, Row-Equivalent Matrices

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3b. Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples

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4. Row-Reduced Echelon Matrices

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40. Norms on Vector spaces. the Gram-Schmidt Procedure I

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41. the Gram-Schmidt Procedure II. the QR Decomposition.

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42. Bessel’s Inequality, Parseval’s Indentity, best Approximation

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43. best Approximation: Least Squares Solutions

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44. Orthogonal Complementary Subspaces, Orthogonal Projections

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45. Projection Theorem. Linear Functionals

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46. the Adjoint Operator

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47. Properties of the Adjoint Operation. Inner Product Space Isomorphism

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48. Unitary Operators

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49. Unitary Operators II. Self-Adjoint Operators I.

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5. Row-Reduced Echelon Matrices and Non-Homogeneous Equations

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50. Self-Adjoint Operators II-Spectral Theorem

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51. Normal Operators-Spectral Theorem

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6. Elementary Matrices, Homogeneous Equations and Non-Homogeneous Equations