This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms filled by students. More often than not, I use the so-called problem method in the courses I teach, and I advocate a particular philosophy of student-centered teaching. Yet, when I evaluate myself, something bothers me. As a professional mathematics educator, the best I can do is to help my students to learn the concepts and the techniques of the course internally, i.e. bounded to the syllabus of the course.
What if I could see beyond the course? What if I was an active mathematician who indeed works with those concepts and techniques, and knows a more advanced and perhaps more general version of those ideas? I was faced with these questions years ago when people started to compare my teaching with the teaching of a mathematician who is indeed an excellent "traditional" lecturer. To my own view, in a sense he could give to his students "more", since he could also see beyond the course. I had forgotten the whole issue until the current term; for the first time I am teaching a course in linear algebra and matrices for mathematics undergraduate students. That excellent colleague of mine is not around now (!), but the question is badly with me:
If I could see beyond the course what ideas (concepts, techniques,
theorems, proofs, problems) would I stress more?
To keep the question suitable for MO, please do not "argue", and just give one piece of concrete advice to a person who now teach to potentially some of your future colleagues!
PS. In this paper (Moore and Less; PRIMUS) you may find the story of the course that the comparison mentioned above started with.
Best Answer
In my opinion, what you should stress in a course on linear algebra depends more on what the particular students in your class want and/or need, and less on what you can "see beyond the course." However, since you asked this on MathOverflow, you are presumably asking for some insight into how professional mathematicians think about linear algebra, so I will try to address that question.
I would say that one the main hallmarks of those who have truly mastered linear algebra is that they can see how linear algebra is applicable in situations where the less well-trained do not. They are able to detect the presence of the "abstract structure" of linear algebra lying under the surface, even when it is not immediately evident from the statement of the problem.
Here are some examples.
Sound waves can be decomposed into a weighted sum of pure tones. "Weighted sum" signals "linear algebra" to the cognoscenti. It doesn't matter that what you're adding together are functions and not finite sequences of numbers, and it doesn't matter that there are infinitely many possible pure tones. What matters is that you can take weighted sums, and that there is a precise sense in which different pure tones are "orthogonal" to each other. That means that linear algebra is applicable, and the concepts of eigenvalues and eigenvectors (or eigenfunctions) are applicable.
The Netflix Prize competition asked for an algorithm to recommend new movies based on your ratings of movies you've already seen. Where's the linear algebra? Start by writing down a large matrix with rows representing people, columns representing movies, and entries representing ratings. Experienced mathematicians know that the biggest singular values of this matrix capture most of the relevant information in it, and provide a good start to constructing the desired algorithm.
An old Putnam problem asked whether two matrices $A$ and $B$ with the property that $ABAB=0$ must also satisfy $BABA=0$. The obvious approach is to start playing around with examples, and there's nothing wrong with that. However, a more insightful approach is to build an abstract vector space with the basis $e_\emptyset, e_A, e_{BA}, e_{ABA}, e_{BABA}$ and define the linear transformations $Ae_S := e_{AS}$ and $Be_S := e_{BS}$, where $S$ is any string of $A$'s and $B$'s, $AS$ and $BS$ denote concatenation, and $e_{S} = 0$ if $S$ is not one of the strings $\emptyset$, $A$, $BA$, $ABA$, $BABA$. This is admittedly a very clever proof and even professional mathematicians might not think of it right away, but this example underlines the power of understanding that anything can be used as the basis of a vector space, even strings of symbols.
Suppose you have a large system of polynomial equations in $x$, $y$, and $z$, containing equations such as $xyz + 4x^2y - z^3 + 7 = 0$ and $y^2z^2 - xyz + 3 = 0$ and many others. At first glance we might think that linear algebra does not help here because we have variables multiplied together, and multiplication is nonlinear. However, if we have enough equations, and if the same terms appear often enough (in this example, $xyz$ appears in both equations), then we might be able to solve the system by using linear algebra, by treating each term as a separate variable and think of the system as a giant system of linear equations in a much larger number of variables. This might seem like a hopelessly optimistic approach, but in fact it is the basis for a general technique for solving systems of polynomial equations. Again, my point is that with a practiced eye, you can learn to see an entity such as $xyz$ not only as a product of three variables, but as a basis vector in a very large vector space.
These examples may not translate directly into useful material for your teaching. However, I do believe that they give a good taste of how mathematicians think about linear algebra. They have internalized what "linear structure" means in the abstract and are able to detect it everywhere, to their advantage. Ideally, one would like to train students to think the same way. Of course, that may be more easily said than done.