How can I learn algebra intuitively

Linear algebra 1

C'est par la logique que l'on prouve
et par l'intuition que l'on découvre.

- With logic we prove
with intuition we discover.
Henri Poincaré (1854-1912)

The Matrix 2020/21
Poster in the fan shop

Lecture in winter semester 2020/21.
Lecturer: Michael Eisermann.
Assistants: Friederike Stoll, Arne Geyer.

We use the learning platform of the University of Stuttgart. Current information on linear algebra, teaching material, forum, dates, etc. can be found on our lovingly designed Iliad page.

Please register for this course as early as possible via C @ mpus / Ilias so that we can reach all participants. Thanks a lot for this!

On this additional, public website you will find:

Ask? For all your questions, organizational or content, please contact Ms. Stoll or Mr. Geyer or me. Feedback? We look forward to your comments and suggestions! Please let us know how you like the event, how you are coping with the material and what can be improved. It is your degree!

Goal setting

Linear algebra and analysis are traditionally the introductory courses in mathematics. They belong to the beginning of the course like learning the letters at the beginning of the elementary school: It is arduous at first, but it is useful for a lifetime! They form the essential foundations on which everything else is based.

Linear methods are immensely practical, so they are used everywhere, both within mathematics and in its many applications. Many calculations eventually lead to matrices and vectors, especially for large amounts of data on the computer. Usually you solve a specific application problem by tracing it back to linear algebra and thus being able to work on it efficiently.

At the same time, linear algebra is also a wonderful theory, which you can use to learn how modern mathematics is structured. The success of mathematics in its numerous practical applications is based on the underlying abstract theory as a stable foundation. Abstract is not a dirty word, on the contrary! Abstract means versatile, flexible and efficient. Theory and practice work wonderfully together, like the left and right hand, only together are they permanently successful.

The lecture Linear Algebra introduces you to this methodical and structural way of working. This results in the learning objectives of the safe handling of vector space structures and linear mappings, matrices and systems of linear equations, the independent solving of mathematical problems, the ability to abstract and mathematical argumentation, as well as precise formulation and writing, and appropriate communication.

Planning of linear algebra in winter term 2020/21

Please continue to inform yourself regularly about the general situation on the website of the University of Stuttgart about current information about the coronavirus. We follow this reliable source of information in all university matters.

The department has put together the most important information for freshmen especially for studying mathematics: questions and answers, links to lecture pages, mailing lists and contacts, last but not least, the mathematics department.

Your LinA team carefully prepares the winter semester. On the one hand, we want to guide you, dear freshmen, as well as possible into your studies and support you in a helpful manner. On the other hand, the corona pandemic calls for great caution. In short, we navigate by sight. We want to offer you a stable framework for the next few months so that you can start your studies well, acquire solid foundations and study successfully. We try to optimally prepare what is currently possible and also to react appropriately to possible changes.

  • Lecture: completely digital as lovingly homemade videos
  • Lecture exercises and open consultation hours: online
  • Exercise groups and consultation hours: online in Webex
  • Café Matrice: weekly in Webex

Information on our offers and materials can be found on Ilias.

Update from Friday, October 30th, 2020, at the beginning of lectures on Monday, November 2nd, 2020: Due to the latest developments in the corona situation, all of our events are exclusively digital. The additionally planned presence offers will be suspended as long as the situation requires it. We regret this last-minute change, but it is urgently needed, so we ask all participants for their understanding and support. We are doing our best to absorb this setback as well as possible with our diverse digital offerings.

My personal experiences are differentially optimistic: All of us, teachers and learners, lack direct contact and mutual exchange. Fortunately, our digital learning platform Ilias has been tried and tested and does everything it needs. Nevertheless, the necessarily digital degree differs noticeably from the actual degree in face-to-face. A lot is easier and more direct when you are present, but we do everything to ensure that it also works well digitally. And some things actually work better digitally.

This winter semester will be a great challenge for all of us. Please do not hesitate to ask many questions; this not only helps you, but also your lecturer. In digital teaching, everyone involved must communicate particularly proactively and carefully. Please do not save with friendly feedback, constructive criticism and praise too: everyone is happy to receive friendly encouragement, especially in troubled times. Stay engaged and patient, even if there is a crunch here and there.

Preparation before the start of the semester

In October you still have some time to prepare for the winter semester. We recommend the preliminary course on mathematics at the MINT-Kolleg. Its aim is to repeat the mathematics material of the school and to bring all participants to a uniform level, but also to give an initial insight into the mathematical content of the course. In addition, it introduces the academic way of working, which differs greatly from the academic way of working and, based on experience, can be problematic at the beginning of the course: large lectures, more freedom, but also a lot of personal responsibility.

Perhaps you know (and love) the Gaussian algorithm for solving systems of linear equations from school. It's wonderfully simple, elegant, and efficient! I realized a long-cherished project and wrote a didactic online tool: Gaël. It's intuitively clickable so you can play with it! You can find a nice application of profit expectations in our seven sample tasks for studying mathematics.

The two overview courses Essence of linear algebra (about 3h, 15 videos) and Essence of calculus (about 3h, 12 videos) by Grant Sanderson in his channel 3Blue1Brown offer a wonderful introduction. The videos are perfectly designed in terms of content and graphics, so wonderful binge-watchable. Take a careful look, it's time well invested!

Zach Star offers two short and beautiful videos on the application of matrices on his YouTube channel: Applications of Matrices (24min) and Applications of Eigenvectors (23min) including an explanation of his animation tools (4min). The latter is freely available on, you can experiment with it yourself.

Literature - my personal recommendation

There are many good textbooks on linear algebra. As a core program, all deal with the classic topics that must be acquired in the first year of every mathematics degree and form the basis for everything else. However, textbooks differ in the breadth or brevity of the presentation, in additional topics, applications or historical insertions, and in the number of examples and exercises.

Please look through several books, skim through the contents, and browse a little as you like the style. You may want a brief, easy presentation to start with, then be ready for a larger, in-depth textbook. You may find numerous examples and applications motivating or distracting. In the end, it is a matter of taste which book you study best with.

To start with, I'll limit myself to five recommendations; a detailed list of literature is more useful in retrospect for connoisseurs. Many of the books not mentioned here would be equally suitable or perhaps even better specifically for you (see above). However, I force myself to do one small ones Choice and allow me my personal preferences. The Stuttgart University Library offers three of the following five groups as eBooks. Widespread textbooks like these can also be bought cheaply second-hand.

  • Gerd Fischer: Learning book for linear algebra and analytical geometry, Springer 2019, 4th edition. I like the new textbook even better than the previous textbook, Gerd Fischer: Lineare Algebra, Springer 2014, 18th edition. There is also a well-coordinated exercise book, Springer 2017, 9th edition.

    Fischer's textbook is probably the most widely used introduction to linear algebra in Germany. Its very successful presentation places great value on motivation and examples, and it also offers many exercises. In my first semester I found the then 250 pages pleasantly short and clear, but later too narrow.

  • Egbert Brieskorn: Lineare Algebra und Analytische Geometrie I, Springer-Vieweg, 1983 and Volume II, Springer-Vieweg, 1985. Volume III, Springer, 2019 was published posthumously. Unfortunately, the three volumes do not appear to be included in our Springer eBook package. Pity!

    This wonderful textbook is unique, as knowledgeable as it is comprehensive, but unfortunately too extensive for most beginners. However, those who have the courage and perseverance will be richly rewarded here! In my first semester, I found Brieskorn's book overwhelming and a little intimidating, but then wonderful.

  • Siegfried Bosch: Lineare Algebra, Springer 2014, 5th edition.

    Compactly written and well structured, with numerous tasks and detailed solutions. Continuation in Siegfried Bosch: Algebra, Springer 2009, 7th edition.

  • Klaus Jänich: Lineare Algebra, Springer 2008, 11th edition.

    Jänich writes wonderfully elegant, everything seems easy and effortless, the depth can only be recognized when you rework it yourself. With 110 test questions for your own exam.

  • Gilbert Strang: Introduction to Linear Algebra, Wellesley-Cambridge 2016, 5th edition. Gilbert Strang: Linear Algebra and Its Applications, Brooks + Cole 2005, 4th edition. Gilbert Strang: Linear Algebra and Learning from Data, Wellesley-Cambridge 2019.

    These are proven linear algebra textbooks with a view to their applications. In the last few decades, the importance of linear algebra has grown in general in numerical mathematics and, most recently, especially in machine learning. The sympathetic lectures are also available on YouTube and MIT OpenCourseWare.

Proper learning

Your studies at a university give you a lot of freedom, that is a good thing. For successful learning, you therefore need a high degree of discipline and self-organization, which is even more true in this digital semester than usual. The Mathematics department currently offers a handout for digital teaching on the information page for students.

  • Try to structure your week and your work day.
  • Keep a logbook and regularly check your learning progress.
  • Form virtual study groups and discuss the content of your studies.
  • Use quizzes and exercises to test and deepen your understanding.
  • Read a textbook to get a different perspective on the subject.

When studying mathematics, it is very important that you form study groups from the very beginning in order to help each other to discuss the material, to exchange ideas and to motivate. In digital teaching, this requires more activity than usual. Please make contact with fellow students early and in various ways, for example when introducing the mathematics department for the first semester. The specialist group is also a good point of contact at any time.

We choose to study Mathematics,
not because it is easy, but because it is hard,
because that goal will serve to organize and
measure the best of our energies and skills.

based on John F. Kennedy (1917–1963)