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I'm going to be taking a semester of math after my bachelor's in economics before I go on to do a master's, and one of the mandatory courses in that semester is linear algebra with a focus on vector geometry. This is how they describe it:

The course gives an introduction to elementary linear algebra with a focus on vector geometry.

Analytic geometry in two and three dimensions: vectors, bases and coordinates, linear dependence, equations of lines and planes, inner product, quadratic curves, calculation of distances and angles, vector and volume product, calculation of area and volume.

Is this stuff useful within economics? I'm fairly sure the other courses are useful but I'm unsure about this one, partly I guess because I don't have a clear picture of what vector geometry really is. The course is not designed specifically for economics students so that's why I'm asking.

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Chisq
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    When comes to financial mathematics there's literally no limit of how advanced maths there is and linear algebra is the foundation of machine learning (together with calculus and probability) which is extremely hot right now (not only in finance, but there also). – mathreadler Oct 07 '17 at 06:15
  • So specifically the vector geometry (which I understand is a subset of linear algebra?) will be something I'm likely to have some use for as an economics/finance professional? – Chisq Oct 07 '17 at 06:17
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    If you want to learn how these new things like trading bots and their algorithms work it will help to learn to crawl before you can walk. Maybe it is not necessary to learn that to get a job.. But I know that if you want to understand that stuff there exists no shortcut avoiding linear algebra. – mathreadler Oct 07 '17 at 06:34
  • Sounds great. Math is the future. – Chisq Oct 07 '17 at 06:36
  • Linear algebra will provide you with simple concrete examples for important mathematical concepts which are outside of e.g. calculus or the other mathematical subjects you know from school. To learn these more abstract concepts its very useful to know the "simple" examples from linear algebra. – Raphael J.F. Berger Oct 07 '17 at 08:23
  • @chisq Math is the past and present as well. – DRF Oct 07 '17 at 12:45
  • You will definitely encounter linear algebra in (advanced) economics. A first course that focuses on the geometry in two or three dimensions probably won't reach the level of abstraction and complexity you will need eventually, but it's a good place to start. – Ethan Bolker Oct 07 '17 at 12:59
  • Linear algebra is almost always relevant whether it is pure or applied mathematics. "Vector geometry" is not a common term in my experience, but it seems to just be a basic linear algebra course. Linear algebra could quite easily be the single most useful mathematical subject both within and outside of mathematics. This definitely applies to economics. – Derek Elkins left SE Oct 07 '17 at 13:01
  • See a related but more general question at the corresponding site: https://economics.stackexchange.com/questions/3304/which-areas-of-mathematical-knowledge-are-required-for-understanding-different-g – FooBar Oct 07 '17 at 14:18

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Econometrics is largely mathematics. If you want to be able to understand it, you have to know the math behind it. Otherwise you will be in a perpetual struggle to grasp the math in what you are learning. So, yes, it is relevant.

MPW
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  • So even the vectory geometry is relevant? It's funny you mention econometrics since that's actually what I'm thinking will be my focus during my master's degree and a part of the reason I wanted to take a semester of math. Very encouraging. – Chisq Oct 07 '17 at 06:19
  • A geometric setting is natural for linear algebra because it helps you understand the mathematical concepts with intuitive examples. Don't dismiss it because you think it's just geometry. It's a stepping stone to more abstract concepts. Linear regression is often expressed in linear algebraic notation, for example. It's surprisingly succinct. – MPW Oct 07 '17 at 06:29
  • Sounds absolutely perfect. Thank you. – Chisq Oct 07 '17 at 06:30
  • And if you have any thoughts of pursuing an actuarial career, you'd better be a competent mathematician. – MPW Oct 07 '17 at 06:31
  • Unfortunately I think actuarial mathematics is a little too advanced for what my master's degree will teach me, my goal is mainly to be a good mathematician "for an economist." – Chisq Oct 07 '17 at 06:35
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A really vivid example is Modern Portfolio Theory (MPT) (Nobel Prize in economics). Linear algebra (or vector geometry) deals with matrices a lot, and these 3 articles show how to deal with MPT (specifically, how to calculate the efficient frontier) using just vectors and matrices

If later you decide to become a quant, this is an example of topics you'd expect to be familiar with (and MPT is there).

rtybase
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As others have pointed out, Linear Algebra is very relevant in economics. I would say that it universally the most useful "higher-level" mathematics (i.e. excluding basic algebra), both in terms of applications and in studying other higher-level math.

But, as you point out, it is not obvious why the focus is on vector geometry. To understand this, you need to know a bit about Linear Algebra. At least at the elementary level, it is a very much a mix of algebra and geometry. However, it can be taught, and often is taught, only from the perspective of algebra. In my experience, this just results in most students being confused by the material, both in terms of understanding the content and its actual purpose/motivation. By teaching the geometry, and getting students to think in terms of the geometry, they actually understand the algebra better.

Now, in this particular case, it is not clear if this is exactly the intent of the class. It seems even somewhat more focused on geometry than I personally think makes sense, so the intent might be different (no way to know without context, of course). Nevertheless, you'll likely find it gives a solid foundation in a very important field of math.

Lastly, I'd like to elaborate on why it is helpful to focus on geometry in studying algebra. As MPW pointed out nicely in the comments, framing things in terms of geometry helps with intuition. That is, while you can and should develop intuition for algebra independently, geometry provides a convenient and deep source of intuition, taken from everyday life. This really helps in understanding the (generally quite abstract) concepts of Linear Algebra. It's always good to use previous knowledge to understand new material, and this is no exception. It's even valuable later when you have the algebraic intuition to see how the two match up and compliment each other.

Jacob Maibach
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I wonder why nobody mentioned (Mixed) Integer Linear Programming (M)ILP. See for example here. I would consider traveling salesman, vehicle routing, shob-shop scheduling and many others as import problems for ecomomics. If you want to understand what's going on, you need vector calculus and optimization techniques.

Christoph
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