Linear Algebra Chapter 6

[Marketta Carucci]

Linearalgebra is the study of linear combinations. It is the study of vector spaces, lines and planes, and some mappings that are required to perform the linear transformations. It includes vectors, matrices and linear functions. It is the study of linear sets of equations and its transformation properties.

All these three concepts are interrelated such that a system of linear equations can be represented using these concepts mathematically. In general terms, vectors are elements that we can add, and linear functions are the functions of vectors that include the addition of vectors

As we know that linear algebra deals with the study of vector spaces and the linear transformations between them. By the definition of vector, it is a physical quantity that has both magnitude and direction. A vector space is defined as the collection of objects called vectors, which may be added together and multiplied (i.e. scaled) by numbers, called scalars. Generally, real numbers are taken as scalars, but there exist vector spaces with scalar multiplication by non-real numbers, i.e. complex numbers, or naturally any field.

The operations such as vector addition and scalar multiplication must satisfy specific requirements, called vector axioms. Generally, the terms real vector space and complex vector space are used to define that the scalars are real or complex numbers, respectively.

An element of a specific vector space may have different characteristics. For example, the elements can be a sequence, a function, a polynomial or a matrix. Linear algebra is affected by those properties of such things that are common or familiar to all vector spaces.

A linear function is an algebraic equation in which each term is either a constant or the product of a constant and a single independent variable of power 1. In linear algebra, vectors are taken while forming linear functions. Some of the examples of the kinds of vectors that can be rephrased in terms of the function of vectors.

Matrices are linear functions of a certain kind. Matrix is the result of organizing information related to certain linear functions. Matrix almost appears in linear algebra because it is the central information of linear algebra.

A room contains x bags and y boxes of fruits and each bag contain 2 apples and 4 bananas and each box contains 6 apples and 8 bananas. There are a total of 20 apples and 28 bananas in the room. Find the value of x and y.

Numerical linear algebra is also known as the applied linear algebra. Applied linear algebra deals with the study of how matrix operations can be used to create computer algorithms, which helps to solve the problems in continuous mathematics with efficiency and accuracy. In numerical linear algebra, many matrix decomposition methods are used to find the solutions for common linear algebraic problems like least-square optimization, locating Eigenvalues, and solving systems of linear equations. Some of the matrix decomposition methods in numerical linear algebra include Eigen decomposition, Single value decomposition, QR factorization and so on.

Currently I am reading "Deep Learning" by Ian Goodfellow, Yoshua Bengio, and Aaron Courville. I'm on Chapter 2 which is the Linear Algebra section where they go over the linear algebra that pertains to the book. I understand most of what is being taught but not at a deep level. And when I get to some of the latter parts of the chapter like "2.9 The Moore-Penrose Pseudoinverse" and specifically "2.12 Example: Principal Components Analysis", I don't really understand them that well at all.

The initial chapters, of this book or any other math book, lay out tools that you will be using in later chapters, so strictly speaking, you will not understand the rest of the book without understanding these foundational chapters.

Realistically speaking, don't worry if you don't understand something. Continue reading until the topic actually appears and is applied. Then, and only then, re-read the earlier section, and try to make sense of it in the light of the later application. By then, you will have seen a lot of other material and may be able to understand it much better against this background.

In addition, it is often very good to look at other sources at this point, when you actually need to understand the application of something. Different authors have different ways of explaining stuff. Looking at things from different angles can be very helpful.

It has been said that good mathematical writing is the kind where you can mentally replace every formula by "foo" and still understand the gist. Read the formulas when you need to understand something in depth and detail.

The Moore-Penrose pseudoinverse is fundamental when you want to create an actual estimation algorithm. If you are mainly interested in applying algorithms someone else has developed and implemented, then you need to understand that algorithm, but much less so the gory details. I have never needed to understand the Moore-Penrose pseudoinverse. We only have very few threads here on it, too.

PCA is much more useful to someone actually applying a tool. Conversely, someone building a tool will likely not use it very much. It's really good to understand this and related ways of reducing dimensionality or compressing information. If you come across a situation where PCA can be helpful in preprocessing, there will not be a big sign pointing this out, so you need to develop your own intuition and understand that this method exists. Happily enough, we have an astronomically upvoted mother-of-all-canonical-threads on PCA, along with an entire pca tag. Go through that thread, then re-read Goodfellow et al. on PCA. Enlightenment is almost sure to follow.

So far the exercises are not particularly challenging. This worries me because I'm not a student at any university and I have no mathematician friends to talk to and so the only way for me to check my level of understanding is doing the exercises.

All in all, I think the book can be a really good place to learn algebra. Obviously this is only my personal opinion, there will certainly be others (probably knowing much more than myself on the subject) with different views on the subject.

Algebraic arguments are generally more elegant, and more enlightening, when one works with arrows rather than elements. Don't worry about the book not being "serious" enough; the whole point of the text is to start you off thinking with the same language as "serious" mathematicians in algebra-heavy disciplines. The main reason to stay away from Aluffi is that category theory is rather abstract, and can seem difficult and/or pointless until one has built up a library of examples. This doesn't seem to be a problem for you, so Aluffi is probably a pretty good choice.

Saal, I second the opinion, based on starting the book, that Aluffi has one of the most user-friendly intros to category theory around. And that is a very good thing because one can relatively easily make one's way through the basics of abstract algebra then hit the wall of categorical thinking and get lost and discouraged. And given its importance as well, it is very nice that you seem to be getting it.

As others as pointed out, if you want more on group representations, you can look at Dummit and Foote, or Lang, and I'm sure Jacobson's Basic Algebra (yes, it has long chapters on Galois Theory in vol. 1 and Rep. Theory in vol. 2). Also, Emil Artin's little book on Galois Theory I recall as being very concise and clear.

Indeed, if you want something that is a bit more 'concrete,' and less concerned with arrows and diagrams, then in increasing order of difficulty, these are excellent books. I am currently concentrating on Dummit and Foote while starting to look at Aluffi for decent insight:

I am a first year undergraduate student majoring in mathematics. Due to the travel ban caused by the Covid-19 outbreak, I am suspending my studies from my uni in the second half of the year (so I have roughly 6-7 months of free time from now). I am planning to teach myself some abstract algebra as well as advanced linear algebra during the break.

More specifically, I am looking for a thorough introduction (and hopefully as general as possible while still being accessible for undergrad) to the basic algebraic objects like groups, rings, fields and vector spaces. The book I am considering to use is Aluffi's Algebra: Chapter 0. This is because the book seems quite comprehensive and self-contained (at least by looking from its table of contents), and the fact that Aluffi introduces categories along the way is also a huge attraction to me that most textbooks don't have.

The thing is, by far I have taken first-year linear algebra, but I have never studied abstract algebra in a systematic manner. I have seen quite a few comments/reviews saying that Aluffi's book is for a second exposure to modern algebra.

So, speaking of my situation, may I ask if Algebra: Chapter 0 can be used as the first book for self-studying abstract algebra (& advanced linear algebra)? And in particular, what would be the pros and cons in doing so?

This book develops linear algebra around matrices. Vector spaces in the abstract are not considered, only vector spaces associated with matrices. This book puts problem solving and an intuitive treatment of theory first, with a proof-oriented approach intended to come in a second course, the same way that calculus is taught. The book's organization is straightforward: Chapter 1 has introductory linear models; Chapter 2 has the basics of matrix algebra; Chapter 3 develops different ways to solve a system of equations; Chapter 4 has applications, and Chapter 5 has vector-space theory associated with matrices and related topics such as pseudoinverses and orthogonalization. Many linear algebra textbooks start immediately with Gaussian elimination, before any matrix algebra. Here we first pose problems in Chapter 1, then develop a mathematical language for representing and recasting the problems in Chapter 2, and then look at ways to solve the problems in Chapter 3-four different solution methods are presented with an analysis of strengths and weaknesses of each.

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