MAT2233

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The Wikipedia summary of Linear Algebra and its history.

Topics List B

Date	Sections from Lay	Sections from Bretscher	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	1.1, 1.2	1.1	Introduction to Linear Systems of Equations	Adding and multiplying equations by constants Solving Equations	Using elimination to find solutions of linear systems The Geometrical interpretation of solutions to linear systems
Week 2	1.3, 1.4, and 1.5	1.2 and 1.3	Vectors, Matrices, and Guass-Jordan Elimination	Introduction to Linear Systems of Equations Equation for a line	Vectors and vector spaces Matrix notation The Guass-Jordan method for solving a linear system of equation The rank of a matrix Sums of Matrices The product Ax (where A is a matrix and x is a vector) The Dot product Linear Combinations
Week 3	1.8 and 1.9	2.1	Introduction to Linear Transformations	Introduction to Linear Systems of Equations Vectors, Matrices, and Guass-Jordan Elimination Transformations of Functions	Linear Transformation Requirements for a transformation to be linear
Week 3	2.1	2.3	Matrix Algebra and Matrix Multiplication	Range of a Function Transformations of Functions	Matrix Operations Matrix products by columns Matrix products using the dot product
Week 4	2.2 and 2.3	2.4	The Inverse of a Linear Transformation	Matrix Algebra and Matrix Multiplication Inverse Functions Introduction to Linear Systems of Equations	The Identity matrix The Inverse of a Matrix The Inverse of a Linear Transformation Various characterizations for an invertible matrix

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	1.1, 1.2	Systems of Linear Equations	Adding and multiplying equations by constants Solving Equations	Vectors and Matrices Gauss-Jordan elimination
Week 2	1.3	Solutions of Linear Systems	Gauss-Jordan elimination Equation for a line	Rank of a matrix Matrix addition The product Ax (where A is a matrix and x is a vector) The Inner product Linear Combinations
Week 3	2.1 and 2.2	Linear Transformations	Range of a Function Transformations of Functions	Linear transformations and their properties Geometry of Linear Transformations (rotations, scalings and projections)
Week 4	2.3 and 2.4	Matrix Products and Inverses	Linear Combinations Inverse Functions Vectors and the Inner product	Matrix Products (both inner product and row-by-column methods) The Inverses of a linear transform
Week 6	3.1	Image and Kernel of a Linear Transform	Solutions of Linear Systems Image of a Function Kernel of a Function	The image of a Linear transformation The kernel of a linear transformation Span of a set of vectors Alternative characterizations of Invertible matrices
Week 6	3.2	Linear Independence	Solutions of Linear Systems Image and Kernel of a Linear Transform	Subspaces of Rn Redundant vectors and linear independence Characterizations of Linear Independence
Week 6	3.2	Bases of Subspaces	Linear Independence The span of a set of vectors	Bases and Linear independence Basis of the image Basis and unique representation
Week 5	3.3	The Dimension of a Subspace	Image of a Function Bases and Linear Independence Linear transformations	Dimension of the Image Rank-nullity theorem Various bases in Rn
Week 7/8	3.4	Similar Matrices and Coordinates	Bases of Subspaces Equivalence Relations	Coordinates in a subspace of Rn Similar matrices Diagonal matrices
Week 9	5.1	Orthogonal Projections and Orthonormal Bases	Parallel and Perpendicular Lines Absolute value function Trig. Functions: Unit Circle Approach Inner Products Bases and Linear Independence	Magnitude (or norm or length) of a vector Unit Vectors Cauchy-Schwarz Inequality Orthonormal vectors Orthogonal complement Orthogonal Projection Orthonormal bases Angle between vectors
Week 10	5.2	Gram-Schmidt Process and QR Factorization	Unit vectors Inner Products Orthonormal Bases Bases and Linear Independence	Gram-Schmidt process QR Factorization
Week 11	5.3	Orthogonal Transformations and Orthogonal Matrices	Image and Kernel of a Linear Transform Inner Products Orthogonal Projections and Orthonormal Bases	Orthogonal Transformations Properties of Othogonal Transformations Transpose of a Matrix The matrix of an Orthogonal Projection
Week 11	5.3	Least Squares	Linear transformations Orthogonal Transformations and Orthogonal Matrices Orthogonal Projections	The Least Squares Solution The Normal Equation Another matrix for an Orthogonal Projection
Week 11	6.1 and 6.2	Determinants	Summation Notation Sgn Function Inverse of a Linear Transformation Transpose of a Matrix	Properties of Determinants Sarrus's Rule Row operations and determinants Invertibility based on the determinant
Week 12	6.3	Cramer's Rule	Determinants Invertible matrices Rotations	Parrallelepipeds in Rn Geometric Interpretation of the Determinant Cramer's rule
Week 13	7.1	Diagonalization	Similar Matrices and Coordinates Orthogonal Transformations and Orthogonal Matrices	Diagonalizable matrices Eigenvalues and eigenvectors Real eigenvalues of orthogonal matrices
Week 14	7.2 and 7.3	Finding Eigenvalues and Eigenvectors	Determinants Invertible matrices Diagonalization Image and Kernel of a Linear Transform	Eigenvalues from the characteristic equation Eigenvalues of Triangular matrices Characteristic Polynomial Eigenspaces and eigenvectors Geometric and algebraic multiplicity Eigenvalues of similar matrices
Week 14	8.1	Symmetric Matrices	Similar Matrices and Coordinates Transpose of a Matrix Eigenvalues and Eigenvectors Algebraic and Geometric Multiplicities	Orthogonally Diagonalizable Matrices Spectral Theorem The real eigenvalues of a symmetric matrix
Week 14	8.2	Quadratic Forms	Symmetric Matrices Finding Eigenvalues and Eigenvectors Conics	Quadratic Forms Diagonalizing a Quadratic Form Definiteness of a Quadratic Form Principal Axes Ellipses and Hyperbolas from Quadratic Forms