Matrix Algebra For Engineers Pdf

[Shanta Plansinis]

CurrentlyI'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What is the difference,if any, between matrix theory and linear algebra?

Let me elaborate a little on what Steve Huntsman is talking about. A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. When you talk about matrices, you're allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another. You can do lots of interesting numerical things with matrices, and these interesting numerical things are very important because matrices show up a lot in engineering and the sciences.

In linear algebra, however, you instead talk about linear transformations, which are not (I cannot emphasize this enough) a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. The difference between a linear transformation and a matrix is not easy to grasp the first time you see it, and most people would be fine with conflating the two points of view. However, when you're given a linear transformation, you're not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you're only allowed to ask for things that don't depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics.

There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.

See, for example Bhatia's "Matrix Analysis" GTM book. For example, doubly-(sub)stochastic matrices arise naturally in the classification of unitarily-invariant norms. They also naturally appear in the study of quantum entanglement, which really has nothing to do with a basis. (In both instances, all sorts of NONarbitrary bases come into play, mainly after the spectral theorem gets applied.)

Sometimes you need concrete computations for which you use the matrix viewpoint. But for conceptual understanding, application to wider contexts and for overall mathematical elegance, the abstract approach of vector spaces and linear transformations is better.

In this second approach you can take over linear algebra to more general settings such as modules over rings(PIDs for instance), functional analysis, homological algebra, representation theory, etc.. All these topics have linear algebra at their heart, or, rather, "is" indeed linear algebra..

I'm with Jon. Matrices don't always appear as linear transformations. Yes, you can look at them as linear transformations, but there are times when it's better not to and study them for their own right. Jon already gave one example. Another example is the theory of positive (semi)definite matrices. They appear naturally as covariance matrices of random vectors. The notions like schur complements appear naturally in a course in matrix theory, but probably not in linear algebra.

This course is all about matrices, and concisely covers the linear algebra that an engineer should know. The mathematics in this course is presented at the level of an advanced high school student, but it is recommended that students take this course after completing a university-level single variable calculus course. There are no derivatives or integrals involved, but students are expected to have a basic level of mathematical maturity. Despite this, anyone interested in learning the basics of matrix algebra is welcome to join.

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This book and accompanying YouTube videolectures is all about matrices, and concisely covers the linear algebra that anengineer should know. We define matrices and how to add and multiplythem, and introduce some special types of matrices. We describe the Gaussian eliminationalgorithm used to solve systems of linear equations and the corresponding LUdecomposition of a matrix. We explain the concept of vector spaces and definethe main vocabulary of linear algebra. Finally,we develop the theory of determinants and use it to solve the eigenvalueproblem.

The course provides an introduction to linear algebra and matrix theory. It is intended primarily for engineering students. This course cannot be used toward the upper level math requirements for MATH/STAT majors. Credit will be granted for only one of the following: MATH 240, MATH 341, or MATH 461.

In order not to intimidate students by a too abstract approach, this textbook on linear algebra is written to be easy to digest by non-mathematicians. It introduces the concepts of vector spaces and mappings between them without dwelling on statements such as theorems and proofs too much. It is also designed to be self-contained, so no other material is required for an understanding of the topics covered.

As the basis for courses on space and atmospheric science, remote sensing, geographic information systems, meteorology, climate and satellite communications at UN-affiliated regional centers, various applications of the formal theory are discussed as well. These include differential equations, statistics, optimization and some engineering-motivated problems in physics.

Contents

Vectors

Matrices

Determinants

Eigenvalues and eigenvectors

Some applications of matrices and determinants

Matrix series and additional properties of matrices

"The book is written in a lovely style: it is easy to read, it is self-contained and assumes no mathematical knowledge beyond high school level. It also contains a huge number of examples showing how linear algebra can be used in other mathematical, physical and engineering domains and even in social science. [...] All in all, the book is one of the nicest elementary books on linear algebra. I recommend it not only for physicists and engineers but also for all students who need linear algebra as a tool." MAA Reviews

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In our last article we covered Linear Algebra introducing matrices and the matrix equation. In this article we will close out our discussion on how to slice and dice our data by applying Linear Algebra thereby introducing matrix algebra. Reader beware, this article is a bit more mathematical than I would like it to be. As a matter of fact, I have done public speaking presentations in the past, around Computational Thinking, and it was received with mixed reviews. Some people tracked with me. But there were some people who did not like the idea of being presented with mathematical facts after a long hard day at work. You will see in our later articles why building foundational maths are essential to Machine Learning. This is certainly one of the prerequisite skills to being a successful Machine Learning Engineer because your model will stop performing at some point in production; think software bugs in normal digitized operations. As a result, your colleagues will be relying on you to fix the model and this does require mathematical intuition among many other prerequisite skills to be successful in the field. OK. I am done with this exhortation. We will now have the benediction. Can I get an Amen?

The last concept that we will discuss is matrix multiplication. It is very similar to regular multiplication with some slight differences. You may remember from grade school the mathematical laws of associative, commutative, and distributive properties. Personally, I have always thought of this as pointless to explain because it seems so obvious. But when it comes to mathematical laws they must be broken down because different branches of mathematics may play by different rules. And Linear Algebra is certainly one of them. We will use scalar variables to explain the basic matrix algebraic laws in order to build up our foundational knowledge. You can think of a scalar like a 1 x 1 matrix. A scalar could be the distance you drove your car in kilometers or how many seconds you can hold your breath under water. It is simply just an individual value that has what is called magnitude or scale with no direction. In terms of Linear Algebra the formal name for a scalar is a called a scalar multiple. We used it in our last article to show how we can make orders of magnitudes of our sweet potato pie!

Going across each row in the first matrix while simultaneously going down each column in the second matrix. Then you multiply each pair together. Then add all of the multiplicative pairs together in each entry. Then put each entry into the new resultant matrix. Go to Step 1 and repeat for the next row and subsequently next column, if any.

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