LADE

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Linear Algebra and Differential Equations

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EN.553.291, Fall 2026

This course is an introduction to the basic concepts of linear algebra, matrix theory, and differential equations that are used widely in modern engineering and science. The discussion will be augmented by topics from linear algebra and the use of computer software (primarily MATLAB).

• Linear Algebra: systems of linear equations, matrices, Reduced Echelon Form (REF), consistent matrices, homogeneous matrices, matrix operations, scalar product, norm, linear independence, matrix inverses, determinants, null spaces, range, spanning subsets, bases, dimension and rank, orthogonal and orthonormal bases, Gram-Schmidt orthogonalization, linear transformations, eigenvalues and eigenvectors, characteristic polynomials, complex eigenvalues, diagonalization.
• Differential Equations: basic DE terminology, separable equations, linear 1st order equations, integrating factors, equilibrium and stability, higher-order differential equations, homogeneous equations with constant coefficients, undetermined coefficients, variation of parameters, numerical solutions, systems of first order differential equations, Laplace transforms.
• Programming: while this will not be our main focus, you will learn how to perform some basic linear algebraic operations with MATLAB, as well as how to solve some differential equations numerically.

Lecture Slides

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Linear Algebra

Matrices and Their Operations Lectures 1–6

Lecture 1

Lecture 1 Systems of Equations

Lecture 2

Lecture 2 Matrices and Row Operations

Lecture 3

Lecture 3 Logic and Set Theory

Lecture 4

Lecture 4 Homogeneous Systems

Lecture 5

Lecture 5 Norms, Dot Product, and Matrix Multiplication

Lecture 6

Lecture 6 Properties of Matrix Operations

Linear Independence and Span Lectures 7–12

Lecture 7

Lecture 7 Linear Independence

Lecture 8

Lecture 8 Nonsingularity and Linear Independence

Lecture 9

Lecture 9 Matrix Inverses

Lecture 10

Lecture 10 Determinants

Lecture 11

Lecture 11 Subspaces

Lecture 12

Lecture 12 Span

Bases and Subspaces Lectures 13–17

Lecture 13

Lecture 13 Bases and Dimension

Lecture 14

Lecture 14 Coordinates

Lecture 15

Lecture 15 Change of Basis

Lecture 16

Lecture 16 Null Space and Range

Lecture 17

Lecture 17 More on Linear Independence and Span

Orthogonality and Rotations Lectures 18–21

Lecture 18

Lecture 18 Orthogonal Based and Gram Schmidt Orthogonalization

Lecture 19

Lecture 19 Linear Transformations and Rotations

Lecture 20

Lecture 20 Orthogonal Matrices

Lecture 21

Lecture 21 Topic TBD

Eigenvalues and Spectral Theory Lectures 22–27

Lecture 22

Lecture 22 Topic TBD

Lecture 23

Lecture 23 Topic TBD

Lecture 24

Lecture 24 Topic TBD

Lecture 25

Lecture 25 Topic TBD

Lecture 26

Lecture 26 Topic TBD

Lecture 27

Lecture 27 Topic TBD

Differential Equations

Linear First-Order ODEs Lectures 28–33

Lecture 28

Lecture 28 Topic TBD

Lecture 29

Lecture 29 Topic TBD

Lecture 30

Lecture 30 Topic TBD

Lecture 31

Lecture 31 Topic TBD

Lecture 32

Lecture 32 Topic TBD

Lecture 33

Lecture 33 Topic TBD

Dynamical Systems and The Laplace Transform Lectures 34–39

Lecture 34

Lecture 34 Topic TBD

Lecture 35

Lecture 35 Topic TBD

Lecture 36

Lecture 36 Topic TBD

Lecture 37

Lecture 37 Topic TBD

Lecture 38

Lecture 38 Topic TBD

Lecture 39

Lecture 39 Topic TBD

Course Demos

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Interactive demos built for this course. They also appear in the main Gallery.

Planes in 3D

Linear Combinations

Parallelogram Law

The Determinant as Area

Visualizing Gaussian Elimination

Gaussian Elimination

Change of Basis

Null Space and Range

Orthogonal Bases and Projections

Gram–Schmidt Orthogonalization