| Week |
Topic |
| 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 |
Matrix Arithmetic, Zero and Identity Matrices, Inverse Matrices, Properties of Powers and Transposes, Elementary Matrix and Row Operations, Systems of Equations and Invertibility |
| 3 |
Simultaneous Solutions of Multiple Systems, Special Matrices (Diagonal, Triangular, Symmetric), Determinants, Inverse of A Matrix by Adjoint Computation, Determinants of Triangular Matrices |
| 4 |
Cramer's rule, Properties of the Determinant, Technique for Determinant Evaluation based on Elementary Row Operations, Determinants of Inverse Matrices |
| 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 |
Euclidean Vector Spaces, Cauchy-Schwartz and Triangle Inequalities, Linear Transformations, Standard Matrices
Midterm 1 (90 minutes- topics until Euclidean Vector Spaces - Euclidean VS not included) |
| 7 |
Reflection, Projection, Rotation, Dilation, Contraction, composition of Linear Transformations, Properties of Linear Transformations, Inverse Transformations. Linearity, Standard Basis Vectors, Standard Matrix in terms of Images of Standard Basis Vectors, Affine Transformations, |
| 8 |
Vector Spaces, Subspace, Examples of Vector Spaces and Subspaces,Homogeneous Linear Systems as Subspaces, Linear Combination- Span- Subspace. Linear Independence, Geometric Consequences of Linear Independence, Functional Spaces and Linearly Independent Functions, Wronskian, Basis and Dimension, The Coordinates Relative to a Basis |
| 9 |
Finite and Infinite Dimensional Vector Spaces, Dimension- Span and Linear Independence, +/- Theorem, Further Theorems, Solution Spaces of Linear & Homogeneous Systems, Rowspace, Columnspace, Nullspace |
| 10 |
Homogeneous, Particular and General Solutions, Determining Bases of Rowspace and Columnspace, Rank, Nullity,Consistency- Solution Space and Rank, Overdetermined and Underdetermined Systems |
| 11 |
Inner Product Spaces, Gram Schmidt Orthonormalization, QR Decomposition, Orthonormal Bases, Least Squares |
| 12 |
Least Squares, Change of Basis
Midterm 2 (90 minutes- topics until Least Squares - LS not included) |
| 13 |
Eigenvalues and Eigenvectors, Diagonalization, Orthogonal Diagonalization
Singular Value Decomposition (time permitting) |
| 14 |
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