Following is the lecture packet written for the [MAT210] Applied Mathematics for Engineers course at METU-NCC by Benjamin Walter. This course was designed by Benjamin Walter, Kursat Aker, and Roald Koudenburg; it is roughly inspired by excerpted sections of G. Strang's graduate textbook "Computational Science and Engineering".
The course teaches engineering students concepts from linear algebra grounded in examples and understanding coming from explicit applications to standard engineering problems. The goal of the course, with a wide range of examples taken from different areas of engineering, is for students to be able to approach (or understand someone else's approach to) new problems in unconsidered applications using linear algebra.
Accompanying the lecture notes are a series of Problem Sheets. The front of the sheet has suggested problems related to the previous notes, and the back of the sheets slowly explain basic concepts from MatLab pertinent to the presented topic.
- Matrices and Matrix Multiplication.
- Matrix Division (Triangular Matrices).
- Problem Sheet: Matrices, Multiplication, and Triangular Division
MatLab: Intro to Matrices - Matrix Division and LU Decomposition (introduction).
- Computing the LU Decomposition of a Matrix.
- Examples Using the LU Decomposition.
- Problem Sheet: Computing and Using the LU Decomposition
MatLab:Editing and Slicing Matrices
- Approximate Solutions, the Normal Equation, and Projection.
- Data Fitting, Orthogonalization, and QR Decomposition.
- Example: Using QR to solve Normal Equation.
- The QR Decomposition as Column Operations.
- Example: Some Scaled Orthogonal Matrices.
- Problem Sheet: Normal Equation, Data Fitting, and the QR Decomposition
MatLab:Normal Equation in MatLab
- Discrete Functions and Euler's Method. ("Vectorizing Functions")
- Discrete Derivatives. ("Vectorizing Derivatives")
- Discrete Impulse Functions.
- Problem Sheet: Discrete Functions, Eulers Method, and 1st Order DE
MatLab:forandwhileloops - Discrete 2nd Derivatives and Boundary Value Problems.
- Discretizing Other Boundary Value Types.
- Discrete Impulse Response and Inverse Matrices.
- Summary of Second Order Discretization.
- Problem Sheet: 2nd Order Boundary Value Problems
MatLab: special matrix creation commands - Extra Note: Error Estimates for Discrete Derivative Formulas.
- MatLab Files
- myLu.m - Example function computing LU decomposition (without row swaps).
- myRef.m - Example function computing echelon form using partial pivoting.
- Euler_method.m - Sample code computing forward Euler.
- Sample Exams
- Truss Systems and Matrices.
- Example Splitting Forces on Truss System.
- Problem Sheet: Truss System Forces and Motions (Linear Approximation)
MatLab:Nulspaces and Echelon Form
- One Dimensional Spring Systems - Equilibrium and Stiffness.
- Computing Spring System Equilibrium.
- Problem Sheet: Spring Systems - Elongation and Equilibrium
(2 pages)
- Intro to Discrete Time Markov Processes.
- Problem Sheet: Introduction to Markov Processes
MatLab: Matrix Powers and Eigenvectors - Computing nth Markov State.
- Problem Sheet: Eigenvalue and Eigenvectors
MatLab: Eigenvalues and Eigenvectors - 1D Spring System Oscillation.
- Falling Systems and MatLab Examples.
- Problem Sheet: Oscillating Spring Systems
MatLab: Functions and Plots
- MatLab Files
- spring_plot.m - Sample MatLab function plotting position of spring system.
- Sample Exams
- Continuous Fourier Series - Facts.
- Continuous Fourier Series - Examples.
- Problem Sheet: Real and Complex Continuous Fourier
- Discrete Complex Fourier Transform.
- (Remark on ω values)
- Roots of Unity and Reduction Formulas.
- (Extra Note: Discretizing Continuous Fourier Transforms).
- Problem Sheet: Discrete Fourier Transform
- Fast Fourier Transform.
- More Fast Fourier Transform.
- Problem Sheet: Fast Fourier Transform
- Infinite and Cyclic Discrete Convolution.
- Filters.
- Problem Sheet: Convolutions and Filters
- MatLab Files
- MATLAB music.m - Sample MatLab code using Fourier transforms to play notes.
- MATLAB voice distortion.m - Sample MatLab code using Fourier transforms to distort an audio recording.
- Sample Exams