ESTR1005 Linear Algebra for Engineers
What is this course about?
This course is an introduction to linear algebra for engineering students. The course covers the following topics:
- Matrices/Vectors
- gaussian elimination
- elementary row operations
- rank/nullity
- vector spaces
- row/column/null spaces
- systems of linear equations
- Determinants: Definitions/Properties
- Cramer's rule
- matrix inverses
- Gauss-Jordan method/determinant method
- linear transformations
- Eigenvalues/Eigenvectors/Applications
- special matrices
- diagonalization
- Basis Transformations/Eigenbasis Transformations
- unitary matrices
- Schur decomposition
- spectral theorem for symmetric/hermitian/normal matrices
- generalized eigenvectors
- Jordan canonical form
- Fields
- Euclid's algorithm
- prime field
- similarities/differences between finite fields and real/complex fields
- Applications of Finite Fields: Codes
- polynomials over finite fields/Schwartz-Zippel lemma/applications
- extension fields
- finite field calculations on wxmaxima
- Sets of Numbers
- limit of a sequence
- Cauchy's convergence criterion
- Bolzano-Weierstrass theorem
- limits of functions
- continuity of functions
- continuity of functions in two variables
- Derivatives
- mean-value theorems
- Taylor's theorem
- Riemann integral
- numerical integration
- Vectors in 2-Space and 3-Space
- definition(s) of dot-products
- properties of dot-products
- more about projections
- vector cross product
- vector cross product/scalar triple product: examples and applications
- curves/surfaces/vector fields
- scalar/vector functions and their derivatives
- lengths of curves
- arc-length parametrization of curves/applications
- Acceleration/Curvature/Torsion
- coriolis acceleration
- multivariate chain-rule/mean-value theorem
- gradient/directional derivatives: definitions/examples
- gradient/directional derivative: proofs
- gradient descent
- multivariate optimization via gradients
- divergence/Laplacian
- curl
- (scalar) line integrals (of vector functions): definitions/basic examples
- line integrals: of vector functions: properties/more examples
- path in/dependence of line integrals: definition/potential function/examples
- path in/dependence of line integrals: proofs regarding potential functions
- double integrals: definitions, basic examples
- double integrals: interchange of order of integration
- change of variables/jacobian: definitions/examples
- surfaces/parametrizations/surface normals/tangent planes
- surface vector integrals
- surface scalar integrals
What is linear algebra?
Linear algebra is the branch of mathematics that deals with vector spaces and linear mappings between these spaces. It is a fundamental part of modern mathematics and has many applications in science and engineering. Linear algebra is used in a wide range of fields, including physics, computer science, economics, and engineering. It is used to solve systems of linear equations, analyze data, and study the properties of geometric objects. Linear algebra is also used in computer graphics, machine learning, and cryptography.