**Week ** |
**Topic ** |

**1** |
1 Matrices, Operations on Matrices, Partitioned Matrices, Matrix Form of A Linear system, Row Echelon and Reduced Row Echelon Forms, Gauss Elimination and Gauss Jordan Elimination, Matrices Defining Transformations |

**2** |
2 Matrix Arithmetic, Zero and Identity Matrices, Inverse Matrices, Properties of Powers and Transposes, Elementary Matrix and Row Operations, Systems of Equations and Invertibility |

**3** |
3 Simultaneous Solutions of Multiple Systems, Special Matrices (Diagonal, Triangular, Symmetric), Determinants, Inverse of A Matrix by Adjoint Computation, Determinants of Triangular Matrices |

**4** |
4 Cramer's rule, Properties of the Determinant, Technique for Determinant Evaluation based on Elementary Row Operations, Determinants of Inverse Matrices |

**5** |
5 Vectors, Arithmetic, Dot Product, Norm, Orthogonal Vectors, Orthogonal Projection, Distance between a Point and a Line, Cross Prouduct, Vector Representation of Lines and Planes, Intersection of Two Planes |

**6** |
6 Euclidean Vector Spaces, Cauchy-Schwartz and Triangle Inequalities, Linear Transformations, Standard Matrices, Reflection, Projection, Rotation, Dilation, Contraction, composition of Linear Transformations, Properties of Linear Transformations, Inverse Transformations, |

**7** |
8 Linearity, Standard Basis Vectors, Standard Matrix in terms of Images of Standard Basis Vectors, Affine Transformations, Vector Spaces, Subspace, Examples of Vector Spaces and Subspaces,Homogeneous Linear Systems as Subspaces, Linear Combination- Span- Subspace |

**8** |
Midterm (120 minutes- topics until Vector Spaces) |

**9** |
9 Linear Independence, Geometric Consequences of Linear Independence, Functional Spaces and Linearly Independent Functions, Wronskian, Basis and Dimension, The Coordinates Relative to a Basis |

**10** |
10 Finite and Infinite Dimensional Vector Spaces, Dimension- Span and Linear Independence, +/- Theorem, Further Theorems, Solution Spaces of Linear & Homogeneous Systems, Rowspace, Columnspace, Nullspace |

**11** |
11 Homogeneous, Particular and General Solutions, Determining Bases of Rowspace and Columnspace, Rank, Nullity,Consistency- Solution Space and Rank, Overdetermined and Underdetermined Systems |

**12** |
12 Inner Product Spaces, Gram Schmidt Orthonormalization, QR Decomposition, Orthonormal Bases, Least Squares |

**13** |
13 Change of Basis, Eigenvalues and Eigenvectors, Diagonalization |

**14** |
Orthogonal Diagonalization, Recitation (time permitting) |