Video and Webinar Series

Differential Equations and Linear Algebra

Massachusetts Institute of Technology professor, Gilbert Strang, explains differential equations and linear algebra which are two crucial subjects in science and engineering.  This video series develops those subjects both separately and together and supplements Gil Strang's textbook on this subject.

    Vector Spaces and Subspaces

    5.1 The Column Space of a Matrix A An m by n matrix A has n columns each in R m . Capturing all combinations Av of these columns gives the column space – a subspace of R m .

    5.4 Independence Basis and Dimension Vectors v 1 to v d are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is a combination of the others. Dimension d = number of basis vectors.

    5.5 The Big Picture of Linear Algebra A matrix produces four subspaces – column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.

    5.6 Graphs A graph has n nodes connected by m edges (other edges can be missing). This is a useful model for the Internet, the brain, pipeline systems, and much more.

    5.6b Incidence Matrices of Graphs The incidence matrix A has a row for every edge, containing -1 and +1 to show the two nodes (two columns of A ) that are connected by that edge.

    Fourier and Laplace Transforms

    8.1 Fourier Series A Fourier series separates a periodic function F(x) into a combination (infinite) of all basis functions cos(nx) and sin(nx) .

    8.1b Examples of Fourier Series Even functions use only cosines (F(–x) = F(x) ) and odd functions use only sines. The coefficients an and bn come from integrals of F(x) cos(nx ) and F(x) sin(nx ).

    8.1c Fourier Series Solution of Laplace's Equation Inside a circle, the solution u (r , θ) combines rn cos(n θ) and rn sin(n θ). The boundary solution combines all entries in a Fourier series to match the boundary conditions.

    8.3 Heat Equation The heat equation ∂u /∂t = ∂2u /∂x2 starts from a temperature distribution u at t = 0 and follows it for t > 0 as it quickly becomes smooth.

    8.4 Wave Equation The wave equation ∂2u /∂t2 = ∂2u /∂x2 shows how waves move along the x axis, starting from a wave shape u (0) and its velocity ∂u /∂t (0).