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# Course Weekly Lecture Plan

 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)