What is the difference between advanced algebra and linear algebra?

Advanced algebra is a general term for the development of algebra to an advanced stage, and it includes many branches. Advanced algebra in universities now generally consists of two parts: linear algebra, preliminary algebra, and polynomial algebra.

Higher algebra further expands the research object on the basis of elementary algebra, introducing many new concepts and quantities that are very different from the usual, such as the most basic ones, such as sets, vectors, and vector spaces. These quantities have the characteristics of operations similar to those of numbers, but the methods of research and operation are more complex.

Linear algebra is a mathematical discipline developed from solving systems of linear equations and discussing the graphs of quadratic equations, and it is a very important basic discipline. Including: determinants, matrices, n-dimensional vectors, systems of linear equations, similarity matrices and quadratic forms, g-vectors, etc.

In terms of course content, most of the advanced algebra is linear algebra, with some polynomial algebra in the middle, and some nonlinear algebra knowledge such as quadratic forms may be taught at the end. Line generation is a non-math major course, and high generation is a math major course. The orientation of the course and the focus of what you have learned are different.

In general, the line generation focuses on the cultivation of computing ability, and can not solve the complex mathematical principles behind it, but the calculation should be accurate and solve practical problems. High algebra, like math scores, is the most basic professional course of mathematics majors, focusing on the training of students' basic mathematical literacy, which not only requires computing ability, but more importantly, understanding the knowledge system and structure, especially the accurate understanding of definitions, the proof of theorems, what are the inferences, etc. The proof of these foundations is often overlooked by the line generation.

In terms of knowledge content, in addition to matrix theory, the core content of the high generation is more focused on the structure theory of linear space and the theory of linear operators, and the latter two parts are not the focus of the linear generation.

High numbers are mainly high numbers, and linear algebra is mainly matrices.

There is a connection. Advanced mathematics and linear algebra are two important branches, advanced mathematics mainly deals with the knowledge of calculus, while linear algebra mainly deals with things from geometry, such as the representation of elements in n-dimensional space is related to the matrix in linear algebra. Calculus in advanced mathematics deals with general tools, and if you want to make a distinction, the only difference is that one is based on calculus and the other is based on matrices.

Mathematics (Hanyu pinyin: shù xué; Greek: English:

Mathematics or maths), whose English is derived from the ancient Greek máthēma), means learning, learning, science. Ancient Greek scholars regarded it as the starting point of philosophy, "the foundation of learning". In addition, there is a narrower and more technical meaning - "mathematical research".

Even within its etymology, its adjective meaning, which is related to learning, is used for exponentialism.

Its plural form in English, and in French with the addition of -es, to mathématiques, which can be traced back to the Latin plural (mathematica), translated by Cicero from the Greek plural ta mathēmatiká).

In ancient China, mathematics was called arithmetic, also known as arithmetic, and finally changed to mathematics. Arithmetic in ancient China is one of the six arts (called "number" in the six arts).

Mathematics originated in the early production activities of human beings, and the ancient Babylonians had accumulated a certain amount of mathematical knowledge since ancient times and could apply practical problems. From the perspective of mathematics itself, their mathematical knowledge is only obtained through observation and experience, and there is no comprehensive conclusion and proof, but it is also necessary to fully affirm their contributions to mathematics.

The knowledge and application of basic mathematics is an integral part of individual and group life. The refinement of its basic concepts can be seen in ancient mathematical texts in ancient Egypt, Mesopotamia, and ancient India. Since then, there has been a steady stream of development.

But algebra and geometry at that time remained independent for a long time.

Algebra is arguably the most widely accepted form of "mathematics". It can be said that since everyone started learning to count when they were young, the first mathematics they came into contact with was algebra. As a discipline that studies "numbers", algebra is also one of the most important components of mathematics.

Geometry was the first branch of mathematics to be studied.

The line generation has little to do with high numbers. High numbers study continuous quantities, while linear algebra studies number arrays, that is, discrete quantities. Specifically, line generation is the study of systems of linear equations, or rather linear transformations in linear space.

Basically, it doesn't matter, it's linear.

To put it bluntly, the knowledge of DU evolved from solving the BAI equation, as long as you are good at learning, you can do it. These two courses are basic courses in mathematics, if you want to use some advanced formulas in the future, then these two courses are the starting bricks, and you have the opportunity to make up for the high numbers!

Again, it's okay to learn directly! It can be said that it is completely new knowledge!

The relationship between the higher and the linear is not too large, the higher is mainly differential and the linear is mainly the matrix determinant or something.

It's a bit related, the focus in line algebra is a bit similar to the idea of solving linear differential equations in high numbers... But not too big.

Linear algebrais a part of higher algebra.

Linear algebra is a branch of mathematics that studies vectors, vector spaces.

Or a system of linear equations for linear spaces, linear transformations, and finite dimensions. Vector space is an important topic in modern mathematics. As a result, linear algebra is widely used in abstract algebra and functional analysis.

Middle. Through analytic geometry, linear generation.

The number can be concretely represented. The theory of linear algebra has been generalized to operator theory. Since nonlinear models in scientific research can often be approximated as linear models, linear algebra is widely used in the natural and social sciences.

Concept:

Linear algebra is a branch of algebra that deals primarily with problems of linear relations. A linear relationship means that the relationship between mathematical objects is expressed in a single form.

For example, in analytic geometry, the equation for a straight line on a plane is a binary linear equation.

The equation for the plane of space is a ternary equation.

A straight line in space, on the other hand, is represented by a system of equations composed of two ternary linear equations, which are regarded as two planes intersecting.

A one-time equation with n unknowns is called a linear equation. A function with respect to a variable that is once is called a linear function. Linear relationship problems are referred to as linear problems. The problem of solving a system of linear equations is the simplest linear problem.

Let's talk about how to learn advanced algebra well

I majored in Mathematics and Applied MathematicsThere are two basic courses in this major, one of which is advanced algebra, and now I have been admitted to graduate school, and advanced algebra is a compulsory subject for mathematics majors, so I have a good understanding of "how to learn advanced algebra?" "I can give this question with more experience. Let's take a look at how to learn advanced algebra well.

My advanced algebra textbook**.

1. Listen carefully to the teacher's explanation in class and take relevant notes

For math students, advanced algebra generally starts in the first year, and if you are just starting out in this course, you will find the content more abstract and difficult to follow and understand by the teacher, which is normal. But no matter what, we must listen carefully when we are in class, follow the teacher's rhythm, take good notes, and try our best to listen carefully if we don't understand, because if we listen carefully, there may be some links that you can still understand, which will be helpful for later learning.

2. Do targeted exercises to reinforce the knowledge points learned

Mathematics is a relatively logical subject, and it is necessary to do some exercises after class. For advanced algebra, the main question types are calculation and proof, and proof questions account for the majority. When we finish learning a chapter of knowledge, we have to do a lot of exercises after class, but this process is not very easy, it is not like high school mathematics, after learning the knowledge points, we will do some corresponding problems, some of the problems in advanced algebra are very abstract, and even some problems require a certain amount of mathematical literacy to solve.

We are based on the exercises in the textbook, and the questions in this book are also very representative, and some of them are postgraduate examination questions. We may only do a few simple calculation questions, proof questions are difficult to write, but we can't give up, we still have to take these questions seriously, complete it, we can buy some reference books online, learn to learn some typical solution methods and ideas, and then do the questions, so that the effect will be better.

3. Learn to summarize knowledge points

After learning a chapter of advanced algebra, we need to spend time after class to summarize the question types and knowledge points that we think are more important in a book, for example, you can use a mind map, the connection between chapters and chapters, and each similar question type is grouped together to summarize the method, which can facilitate review, and make you more familiar with the knowledge points, and the learning process will not be messy because there are too many knowledge points.

Difference Between Higher Algebra and Linear Algebra

There is a difference in content

Higher algebra includes linear algebra, which contains chapters on polynomials, Euclidean spaces, bilinear spaces, and symplectic spaces, while linear algebra does not. With the same knowledge points, the content of advanced algebra is more comprehensive, and the knowledge learned is deeper and more difficult, while the knowledge points of linear algebra are relatively shallow and simpler.

Applicable students are different

Advanced algebra is generally suitable for students studying mathematics departments, while linear algebra is suitable for students who are not majoring in mathematics.

Mathematics is a difficult subject, and there is no shortcut to learning mathematics.

1. How should I learn advanced algebra to learn it well?

Build a good foundation in mathematics.

If you want to learn advanced algebra well, you must first start with books. Conscientiously study every theory and formula in the book thoroughly, check the information when you encounter difficulties, and ensure that you understand the theoretical knowledge and use the formula to solve the problem. If you don't know how to read books yourself, then you should go to the most basic online courses.

After learning the theoretical knowledge, do all the example problems and after-class exercises in the textbook, and when you do it for the first time, you must do it in detail, and you must learn the questions you don't understand clearly. After the first pass, start doing the second and third passes.

Taking notes. Be sure to take notes, but taking notes is not copying. When taking notes, write down the questions that you didn't understand in your own words, so that you can understand them when you review your notes.

In the notes, it is necessary to summarize the methods of doing a type of question. Write down the questions and problems that are easy to make mistakes in a notebook. You can use some tools such as highlighters, post-it notes, etc.

Use your notes to fill in the gaps.

Review in a timely manner. After learning a knowledge point, you must review while the iron is hot in the evening, read more textbooks, and understand theoretical knowledge. Also review the important notes you took and the wrong questions.

Do more questions. To learn mathematics, you have to practice. Whether it's an example problem in a book or an exercise in a workbook, do it over and over again.

Doing more questions does not mean doing a lot of questions, but doing the questions in the book and the workbook over and over again until you don't make any mistakes. Cultivate mathematical problem-solving thinking over and over again to ensure that you can use each knowledge point to solve problems.

2. The difference between higher algebra and linear algebra.

The knowledge of advanced algebra is very detailed, theoretical, much more difficult than linear algebra, and the main learners are mathematics majors.

Linear algebra is a part of advanced algebra, which is a simple course for non-math majors in science and engineering.

All in all, whether it is advanced algebra or linear algebra, you must work hard and diligently to learn well.

I participated in the school's Math Honors Program when I was a freshmanI didn't learn high math and line generation, but I learned math and high generation, and in the end, through my own efforts, I was excellent for two semestersNext, I would like to share my experience of learning from the high generation.

The line generation is only a part of the high generation

Some people have always asked me, what is the difference between the high generation and the line generation? Aren't both learned matrices and determinant knowledge? Actually, it's notIn fact, the content and difficulty of the high generation are greater than that of the line generation

- In other words, Gao Dai is an online substitute class for mathematics students.

Advanced Algebra, focusing on linear spaces on general domains, linear transformations, matrix determinants, and so on. From my personal learning experience,The concept is more abstractSometimes it takes a long time to read the definition, let alone use it in the exam. The content of the matrix determinant in the high generation is exactly the same as that of the line generation, but the scope of the discussion is broader.

(The following is a part of the learning content that the high generation has more than the line generation).

WhileLinear algebraThe focus is on more specific concepts such as determinants, matrices and their transformations, systems of linear equations, quadratic forms, and moreEmphasis on calculations, not proof.

Since it's very difficult, how can you learn it well?

Read the book over and over again + brush up on the questions

This is just my learning method, not necessarily informative. When I was studying Gao Dai, I learned it through the three steps of reading books, thinking independently, and brushing questions. If the book is really hard to understand and I don't understand the class, I recommend reading it"Advanced Algebra" by Professor Qiu Weisheng of Tsinghua University, basically can help you understand the concept by hand.

As long as you think through the textbook knowledge points and understand themBrush up on more calculations(e.g., finding full-rank matrices, orthogonal matrices, quadratic forms, etc.).There are also some proof questions(refers to the kind of questions that have rules and routines to speak of), for some other jerky and difficult proof questions, you can only memorize the routines.

——My personal opinion is that not being able to do the questions is a hindrance to learning from the high generation, so it is still necessary to brush the questions, don't be too troublesome(If you're not after the good or the good, I didn't say).

All in allAlthough the content of the high generation is more abstract and more difficult than the line generation, if you can understand the basic concepts in the textbook well and do more similar questions to consolidate, I believe that excellent has long been in the bag.