Applied Mathematics in Bioengineering

Overview

Covers and emphasizes practical applications of linear algebra, statistics, complex analysis, and signal processing in bioengineering. Students develop fluency with matrix decompositions, statistical inference and curve fitting, complex-number representations of signals, linear time-invariant systems, and Fourier analysis, and apply these tools to representative biomedical problems.

Syllabus

Introductory Linear Algebra

In this module, you will work hands-on with the objects of linear algebra — scalars, vectors, matrices, and tensors — and the arithmetic that connects them. You will represent biomedical data as vectors and feature lists, encode grayscale and RGB images as matrices and tensors, and build affine transformations that scale, rotate, and translate them. Along the way you will assess linear independence, compute matrix rank, and develop the fluency with inner products, matrix multiplication, and inversion that the SVD and PCA work in Module 4 will depend on.

Linear Algebra Fundamentals

In this module, you will learn the language of linear algebra that underlies imaging, signal processing, and biomedical data analysis throughout the rest of the course. You will work with bases and coordinate representations, move data between coordinate systems through change-of-basis transformations, and solve systems of linear equations using Gaussian elimination. The module closes with eigenvalues and eigenvectors, building the foundation you will need for SVD and PCA work.

Basic Statistics and Data Fitting

In this module, you will build the statistical fluency every bioengineer uses to summarize experimental data, fit models to it, and decide whether observed effects are real. You will describe distributions with means, variances, quantiles, PDFs, and CDFs, then move to least-squares regression and interpret residuals, R², and confidence intervals. The module closes with formal hypothesis testing — Student's t-test and one-way ANOVA — and asks you to choose the right test for a given experimental design.

Intermediate Linear Algebra and Statistics

In this module, you will bring together the linear algebra of Modules 1–2 and the statistics of Module 3 to build the dimensionality-reduction toolkit at the heart of modern bioengineering data analysis. You will compute the Singular Value Decomposition of a matrix, interpret U, Σ, and Vᵀ geometrically, and use the dyadic form to construct low-rank approximations. From there you will perform Principal Component Analysis by both the covariance and SVD routes, contrast it with Independent Component Analysis, and decide which is appropriate for a given dataset.