Linear Algebra for Engineers

The following presentation builds on previous posts on linear algebra to establish the use of linear algebra in engineering. Building on the previous articles, this presentation includes a general summary of applications along with direct computation for solving engineering problems. The computational aspect covers general constructions and specific maths for simulation. The presentation also covers the meaning of linear algebra vocabulary in an engineering context.

Contributor: Joseph Brewster

Introduction to Proofs

Proofs are, indeed, a difficult thing in mathematics. But mathematics is at its core just proofs, and less importantly calculations.

To prove requires a certain ingenuity to construct the right object or synthesize ideas in the right order and then show that that proves what you think it does.

For example, prove that the interval \([0,1]\) and \([0,1)\) have the same number of elements. This means you can find a one-to-one function (a bijection) between them. Try it, see if you can figure it out.

I wrestled with this problem for a long time. I tried drawing graphs but they all had a curve, meaning they fail the horizontal line test.

It turns out, that you have to construct a clever object:

\(f:[0,1]\to [0,1)\) by the following:

For all fractions \(\frac{1}{n}, n\in\mathbb{N}\):

\(f:\frac{1}{n}\mapsto \frac{1}{n+1}\)

for all else:

\(f:x\mapsto x\)

You may have heard of this, it’s commonly called Gilbert’s Hotel. In order to map between infinities where one of them has “extra” elements. You shift \(1\) to \(\frac{1}{2}\), \(\frac{1}{2}\) to \(\frac{1}{3}\), and so on. This is indeed a bijection.

Being exposed to problems and solutions like this increases your ability to solve similar problems. Especially in fields like analysis, we require increasingly more and more machinery and ingenuity to solve difficult problems. To prove something new requires even more of the same.

What it’s like to be a PhD student of B. Halstead

Apparently we have some readers in the community of folks with whom Mr. Tetra Halstead associates. Good morning to you all.

Every other school day from 1:30–3:00, the four of us (J. Brewster, B. Halstead, D. Staudinger, D. A-M. Trần) meet in our school’s library or other sufficiently mathematical location and do maths. Previously, we had been going through coursework altogether, under the careful guidance of Mr. Halstead. During that time we worked on introductory Abstract Algebra (Algebra: Chapter 0 by Paolo Aluffi), Real Analysis (Understanding Analysis by Stephen Abbot), and Topology (Topology by James R. Munkres). As of now, we have broken off into working on our own topics which interest us.

I, D. Staudinger, have been occupying lots of Mr. Halstead’s time working on a variety of topics. I have worked a bit on Differential Geometry/Topology and Analysis. Mr. Brewster, being an engineer, has been working more with Linear Algebra and its application with the approximation of bridges and heat transfer and such topics. Mr. Trần has been working with topics in logic and algebra.

Our goal with this independent study is about learning the maths that we are interested in and maybe applicable in our future. I, D. Staudinger, plan to study pure maths in college. Mr. Trần has had a long-standing fascination with logic. Mr. Brewster, being an engineer, is using this time to hone the mathematical basis for simulations and the underpinning of lots of real engineering calculations.

The blog will be updated shortly when we decide where we will all go to college.

Intro to Real Analysis, Cauchy sequences, and the definition of the Real Numbers

In order to any kind of analysis we create a metric space. A metric space is a set equipped with a distance function \(d(a,b)\to [0,\infty)\). We read this as “the distance between \(a\) and \(b\)”. This must satisfy \(d(a,b)=0 \Longleftrightarrow a=b\) and \(d(a,b)=d(b,a)\) as well as \(d(a,b)\leq d(a,c)+d(c,b)\). The first two conditions must clearly be true if we are to call this a distance function, but the last is less obvious. This is called the triangle inequality and it is used a lot. It basically means that our distance function finds the shortest path, and it must be the same length or more to go through a third point. This is obviously true if you think about it in 2d space:

We will now define what makes a sequence converge. Consider the sequence \(\{p_n\}\) which is the set \(\{p_1,p_2,p_3,\cdots\}\). We claim that this sequence converges to \(L\) (limit). Then for any \(\epsilon >0\), there exists an \(N\) such that for any \(n\geq N\), \(d(p_n,L)<\epsilon\).

This formalizes the idea of getting closer and closer to a value \(L\). Any distance \(\epsilon\) I give you, you can give me an index \(N\) after which everything is closer than \(\epsilon\). If I tell you \(\epsilon = 0.001\), you have to be able to give me a number \(N\), perhaps \(N=1000\), so that everything after that is closer than \(0.001\), meaning \(d(p_{1000},L)<.001\) and \(d(p_{45001}, L)0\), there exists \(N\) such that for any \(n,m\geq N\), \(d(p_n,p_m)<\epsilon\). It is important to distinguish this from the definition of convergence. Cauchy sequences have a special relationship with convergent sequences.

It can be proved that convergent sequences are Cauchy. This makes sense, since if elements of the sequence are getting closer to some limit, then they must get closer to each other. However, it is not true that Cauchy sequences all converge. How can this be? Well, I showed an example before, where the “limit” was not in the metric space. In a sense, the metric space is “missing” the value where the limit should be.

In some metric spaces, every Cauchy sequence converges. This is called “the Cauchy Criterion”. For example, in the metric space \([0,1]\), every Cauchy sequence converges. You’ve patched the holes where the limits should be by putting a limit there. In a sense, you’ve completed it. If a metric space passes the Cauchy Criterion, then it is complete. There are no fake limits which are missing from the space.

This concept is very powerful. Consider \(\mathbb{Q}\), the rational numbers. Are they complete? Consider the sequence \(\{3,3.1,3.14,3.141,3.1415,\cdots\}\) which only has rational elements. We know that this will eventually “converge” to \(\pi\), but \(\pi\) is not in \(\mathbb{Q}\) so the sequence does not converge at all.

In a sense we can define \(\mathbb{R}\), the real numbers, as the completion of the rationals. We can define it as the smallest metric space containing \(\mathbb{Q}\) for which the Cauchy Criterion is satisfied. We add in all the points which are the limit of some rational sequence. There are different ways to define the real numbers which I may discuss in the future, but this is a particularly neat one, I think.

Matrices in Calculus

Examples are a very important tool in understanding mathematics, including linear maps and matrices. Common examples of linear maps include differentiation and integration where c = 0 as maps between vector spaces of polynomials. Through experience with these operations, it is clear to see that each follows the properties of additivity and homogeneity. Let D represent differentiation and I represent integration, the properties mean that D(a+b) = D(a) + D(b) and cD(a) = D(ca) for a constant c.

For some degree polynomial, n, the derivative maps \(x^n\) to \(n*x^{n-1}\) and the integral maps \(x^n\) to \(\frac{x^{n+1}}{n+1}\). This way of expressing the operations draws the connection to the basis elements of the space of polynomials. Since we have already defined the maps on the basis elements, a matrix can be constructed for both operations. The general construction of these matrices will be the coefficients (1, 2, 3 … and 1, ½, ⅓…) along the diagonal to represent multiplying or dividing by the power shifted to represent the change in power.
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These matrices, or at least the infinitely large varieties, can be applied to any polynomial to yield the derivative or integral. The way this would be done with matrix operations is by multiplying the matrix for the operation by the n-by-1 matrix representing the input function.

Since the integral and derivative are inverse functions of one another, the composition of the functions yields the original function, meaning that the product of the matrices representing integration and differentiation will be the identity matrix, as seen below.

This example is a good way to understand the notion of how linear maps are represented as matrices because of experience with derivatives and integrals already relying on the necessary properties of linear maps.

Another matrix in calculus is the total derivative of a function, which is a linear map from \(R^n\) to \(R^m\) based on a function between the two sets. The total derivative of a function is a set of linear maps between the basis elements of the domain and the rate of change of a particular element of the basis of the codomain in that direction. Your text to link here…

Under the condition that all the partial derivatives exist, the total derivative corresponds to the Jacobian matrix, which is used to transform spaces in vector calculus.

The exact transformation created by the Jacobian matrix is left as an important exercise to those studying vector calculus.

Contributor: Joseph Brewster