Where will engineers use vector spaces ?

Hey All,

I am tutoring a mixed class of (mostly) engineers and physicists and I am trying to get across how important the concept of a vector space is - that its not just some abstract problem that only pure mathematicians need to worry about.

Its easy to highlight the need for linear algebra for physicists - Quantum Mechanics is entirely based on it. I am finding it harder to justify to engineers where they will use vector spaces - I know its important for time domain (state space) control theory and you describe stresses in materials using tensors.

Can anyone think of other applications of vector spaces for engineers ?

Cheers,
Thrillhouse

I'm studying electrical engineering. In circuit theory, we use matrices to solve for current or voltage. In electromagnetic field theory which is a fundamental course for communication engineering, conception of divergence, curl are important.
For other fields of engineering, computer memory extensively uses the conception of partition of matrices. If the matrices size gets larger than the space of computer memory it divides the matrices into submatrices and does calculation.
Again linear operator plays a key role in computer graphics. For many CAD software generates drawing using linear operators.
And don't forget about cryptography. Matrices can be cleverly used in cryptography. Exchanging secret information using matrix is very robust and easy in one sense.
How about MATLAB? This software is widely used in engineering fields and MATLAB's default data type is matrix.

HallsofIvy
Homework Helper
And, of course, Linear Algebra is the underlying theory for all of linear differential equations.

jasonRF
Gold Member
I am an electrical engineer, and find that vector spaces and matrix algebra come up often. Examples:

1) Least square estimation has a nice subspace interpretation. Many linear algebra texts show this. This kind of estimation is used a lot in digital filter design, tracking (Kalman filters), control systems, etc.

2) FOURIER ANALYSIS! The discrete fourier transform is a nice finite dimensional example, and the FFT algorithm is just fun to learn about. Continuous time is nice too, but then you are in infinite dimensional space of course ...

3) incidence matrices from graphs that represent circuit topology. The kirchof voltage and current laws can then be nicely represented in matrix form. Yes, I had an undergrad electrical engineering class that covered this stuff, and included state-space analysis, matrix exponentials, etc.

4) We use orthogonal projections all the time. For example, in adaptive beamforming, if the interference signals have a very high signal to noise, we essentially project the data orthogonal to the interference subspace in order to maximize the signal to noise of the desired signals. In the limit of infinite interference to noise, you get exactly the subspace projection.

5) Solving nonhomogeneous PDEs using eigenfunction expansions. I know, this is infinite dimension again, but relating sturm-liouville to symmetric matrices, and solving Ax=c by eigenvector expansions is fun. This kind of problem comes up in electrodynamics (electrical engineering), fluids (mechanical/civil/chemical engr.), quantum mechanics (electrical/materials/chemical engr). etc.

6) ODEs, of course. Basic signals and systems courses are basically based on the fact that complex exponentials are the eigenfunctions of constant coefficient ODEs. Fourier transform is basically a projection onto this space.

7) SVD is used everywhere for things like compressing images, decomposing 2-D filters into simple outer products of 1-D filters (much more efficient to implement). SVD for numerics is also important ...

jason
jason

EDIT: the book "linear algebra and its applications" by Strang has nice examples that relate to engineering, included a bunch of those above. The "applications" edition of Anton's "elementary linear algebra" book has a bunch of chapters with nice applications.

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Integral transforms made sense in an entirely new way to me, once I understood them in terms of generalizing vector spaces to an infinite dimensional limit.