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When I was a freshman, I have learnt linear algebra for two semester. I feel linear algebra is abstract and hard to truly understood. For example, my textbook introduce the concept "nonsigular" by introduce the concept "determinant" first. I know how to calculate determinant, but I do not know why we need the concept "determinant" first, what's the geometrical meaning of "determinant", why when determinant is equal to zero so that the matrix is nonsigular.

There are many concepts like "determinant" or result that is easy to calculate or easy to mathematically prove but really hard to understand intuitively or geometrically.

My question:

  1. Why freshmen feel linear algebra is abstract?
  2. Should we learn or think geometrical meaning of every basic concept in linear algebra? Does intuitive important to linear algebra?
  3. What should a freshman or one wants to understnad linear algebra better do?

Update:

  1. "determinant" is just a example.
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In Euclidean analytic geometry, the determinant of a square matrix is the signed volume of a parallelepiped spanned by the columns(or the rows) of the matrix. So if the determinant is $0$, the parallelepiped collapses. Hence the matrix is called singular. –  Makoto Kato Nov 25 '12 at 5:38
    
@MakotoKato I doubt Teng will think that will help because I am sure if he doesn't know what a determinant is then he definitely won't know what a parallelepipe that collapses is. I sure don't. –  Q.matin Nov 25 '12 at 8:14
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A parallelepiped is like a rectangular prism (in $n$ dimensions) but pinched on all sides, in the sense that a parallelogram is a pinched rectangle. More firmly, it is a polytope ($n$-dimensional polyhedron) with the property that all of its "faces" are lower dimensional parallelepipeds - so a 3D parallelepiped has all its faces as parallelograms. "Collapsing" is a colorful image for saying that one "side" has length 0. You can imagine one face of a cube falling into an adjacent one, which is a visual image of one of the sides going to zero. –  Eric Stucky Nov 25 '12 at 8:21
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@Q.matin Geometrically, a determinant is a scaling factor. If we regard a matrix $A$ as a linear mapping, then $A$ maps the unit cube to another shape, a parallelepiped (a tilted and skewed cube if you will). In the course of this mapping, the volume is changed. The determinant is the scaling factor for this volume change. If the resulting parallelepiped is collapsed (degenerate, imagine a cube collapsed into a square) then the volume (and hence determinant) is $0$. But then this means the mapping is not one-to-one and hence the matrix is non-invertible/singular. –  EuYu Nov 25 '12 at 8:25
    
Mostly because it is. –  Mariano Suárez-Alvarez Nov 25 '12 at 9:04
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3 Answers

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I am surprised both by the approach of your textbook (you don't need determinants to introduce the distinction between singlar and non-singular matrices, nor to solve linear systems), and by the fact that you qualify this approach as abstract. I would qualify a don't-ask-questions-just-compute attitude as concrete rather than abstract. Maybe you use "abstract" to mean "hard to grasp", but it is not the same thing; for me often the things hardest to grasp are complicated but very concrete systems (in biochemistry for instance). In mathematics (and elsewhere, I suppose) it is often asking conceptual questions that leads to abstraction, and I sense that what you would like is a more conceptual, and therefore more abstract approach.

But abstraction is present in many fields of mathematics, like linear algebra, for a more improtant reason as well, namely for the sake of economy and generality. Linear algebra arose as a set of common techniques that apply to problems in very diverse areas of mathematics, and only by an abstract formulation can one express them in such a way that they can be applied whereever needed, without having to reformulate them in each concrete situation. It would be motivating to have seen at least one such concrete application area before entering the abstraction of the subject, and I think that would be a sound approach. However this would involve introducing many details that in the end are independent of the methods of linear algebra, and I guess there is often just not the time to go into such preparations.

So to answer your questions.

  1. Linear algebra is an abstract subject, so it should not surprise tht freshmen feel it is so. But it is not abstract because of determinants, which are just a concrete tool that allows certain things to be expressed more explicitly than without them. Saying a linear map is invertible is a more abstract formulation then saying $\det A\neq0$ where $A$ is a matrix of that linear map in some basis.

  2. Yes, geometric insight helps understanding linear algebra, and you should have some geomtric intuition for notions as subspaces, span, kernels, images, eigenvalues. But determinants are somewhat different; while you certainly should have some geometric intuition for determinants in terms of volume when doing calculus, there is not much to gain from this in purely algebraic situations, and in fact I would know no geometric interpretation at all of the determinant of a complex matrix, or of the determinant that defines the characterisitic polynomial.

  3. To understand linear algebra better, you should try to go beyond concrete computational questions, and try obtain a more conceptual understanding of what is being done.

As for the mysteries of determinants, you may want to have a deeper understanding than just that they exists and magically solve certain problems (like determining which square matrices are invertible). For that I would refer to this question.

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For a linear algebra course intended as a pre-proofwriting exposure to life beyond calculus, I think that considering singularity as fundamentally a statement about determinants makes a lot of sense. Without a lot of training in mathematics, statements like "matrices which aren't invertible are singular" are unsatisfying, because we're not used to definitions being so opaque. Whereas to say that a matrix with zero determinant is singular feels at least graspable, because the determinant is (relatively) easily computable. –  Eric Stucky Nov 25 '12 at 8:17
    
(I should say that I only have experience from the student perspective in this matter) –  Eric Stucky Nov 25 '12 at 8:34
    
@EricStucky I think that approach is a bit flawed. One of the key reasons we care about determinants is because they tell us when a matrix is non-invertible. Defining "singular matrices" as matrices with determinant zero is great, but there is really not much content behind it. Why do we care if the determinant is zero? We care because it is a useful criterion for determining non-invertibility. Ultimately, you can't avoid the issue. –  EuYu Nov 25 '12 at 8:41
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I think this depends on both the student and the professor/lecturer, so the answer to your first question is certainly very person-specific. I can give some suggestions on how to make it less abstract. There are numerous applications of various concepts in LA in many other areas, e.g. Markov Chains (eigenvalues), Statistics/Econometrics (linear models), Operations Research (orthogonality). I recon this is the best point to start.

Specifically, stationary distribution in an MC is an easy and intuitive concept that makes use of matrix inverse and solution to the set of linear equations. It is easy to use a case, e.g. from population modeling where each state of an MC is the number of the infected species in the population and the vector of stationary distribution is the limiting probability to observe this number. This vector is derived by solving the set of linear equations.

There are plenty of examples like this.

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The way I personally like it is to carry the course along the topic of invertibility of matrices. You start with linear systems of equations, you write them better in terms of matrices, and get to the need of inverting a matrix. This leads into determinants, which allow you to decide on invertibility and are necessary to find a formula for the inverse. But then determinants are hard to compute so you end up solving systems by row reduction and this also gives you a way to calculate inverses. In discussing the number of solutions of a system, you get into discussing the rank of a matrix and linear independence.

I am not sure if freshmen see linear algebra as abstract. Rather, I think that the problem is the number of new notions and, more than that, a new way of thinking. My experience is also that students come from highschool trained into math of the form "memorize a formula or a method and apply it in a straightforward way", and struggle when understanding is required.

Regarding geometrical meaning, I'm not sure if it is that important. It is good if it is there, but it might not be easy to teach/learn.

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