Does seeing beyond the course you teach matter? The case of linear algebra and matrices 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.  
 A: My approach to teaching Linear Algebra is primarily geometric, drawing on the way that we utilize linear algebraic techniques in low dimensional topology when computing and understanding knots and $3$--manifolds. I fell in love with a book which teaches Linear Algebra in just this way:

Shafarevich, Igor R.; Remizov, Alexey O., Linear algebra and geometry. Translated from the Russian by David Kramer and Lena Nekludova, Berlin: Springer (ISBN 978-3-642-30993-9/hbk; 978-3-642-30994-6/ebook). xxi, 526 p. (2013). ZBL1256.15001.

It's a bit high-tech for the average freshman Linear Algebra course (although it could certainly be used for a `topics' course), but it's a very good book for the instructor to be looking at alongside the course textbook, in order to add flesh to the skeleton syllabus of the course.
Summarizing, if an instructor can see beyond the freshman Linear Algebra syllabus, then that allows them to provide geometric intuition and to "draw pictures", connected linear algebra with the visual part of the brain and with the way that Linear Algebra is actually thought of and used in many applications.
A: If the question is centered on linear algebra, I personally feel that while from "inner" reasons I could very well survive without teaching the dual vector space and linear forms (which, generally speaking, seem to confuse my students, more than helping them). 
On the other hand I feel that in a longer perspective this topic is extremely relevant: I find the way it is treated in the book by Vinberg "A course in algebra" (GSM 56 of the AMS) very stimulating. 
A: 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.
A: Read up on some mathematical field you find exciting. It is bound to involve linear algebra in some form, you will see some aspects of your course in a new light, and the inspiration will be infectious.  Next year, try another field.
A: Very much anecdotal but, for what is worth: 
What I did last time I taught our advanced linear algebra course was to make it a topics course, and develop the theory as needed for the examples I chose to cover. It was very much an experiment (and my attempt to keep the course fresh and interesting for me). 
We proved the fundamental theorem of algebra using linear algebra, saw some applications to number theory (finite fields; there are also nice applications to the theory of error-correcting codes, but we ran out of space before we could cover them), some to graph theory (Algebraic graph theory), some results on numerical approximations to eigenvectors (which allowed me to discuss the Google search algorithm), etc. 
Next term I'll be teaching the course again, and to keep things different, this time  I'll be using as text Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra, by Jiří Matoušek. You may want to look at it. Here is a course page by Donald Allen on linear algebra for applications, with nice lecture notes (that I also used as reference last time). 
A: The thing about linear algebra is that every single piece of it is vital for everyone who is doing modern mathematics, including the stuff that is normally left out of all but the most advanced courses (tensors, determinants, block matrices, various decompositions), excluding maybe the properties of Hermitian matrices if one is doing pure algebra and the rational canonical form if one is doing pure analysis. There is a lot that can be added to a linear algebra class beyond the standard material to provide context, additional generality, new exercises and project topics, but it is not usually worth throwing out the important stuff. I think the problem should be solved by extending undergraduate linear algebra to 3 semesters, but that might be easier to suggest than to realize...
From an advanced perspective, one can work with modules over rings instead of vector spaces over fields. I actually believe this is a good idea (not many agree), because it makes one appreciate (1) the usefulness of division (as seen in the fact that the Gaussian algorithm, and with it most matrix algorithms, breaks over non-fields) and (2) the miraculousness of determinants (for they allow one to salvage at least some linear algebra in the non-field setting). And of course, this pays off when doing various canonical/normal forms (and even just characteristic polynomials), since one no longer needs to do ad-hoc definitions of "matrices with polynomial entries" or "polynomials with matrix coefficients" but just can speak of matrices over polynomial rings.
Andres Caicedo gave a great selection of sources for applications of linear algebra that (often) can be used in classes to provide motivation or exercises. There are also applications to enumerative combinatorics via the Lindström-Gessel-Viennot lemma, allowing one to enumerate various tilings, such as lozenge tilings of a hexagon (the answer contains no determinants, yet they are useful in the proof!). I can provide references if needed. For more advanced-linear-algebra material (although not very organized), see https://math.stackexchange.com/questions/433858/high-level-linear-algebra-book , http://diendantoanhoc.net/forum/index.php?app=core&module=attach&section=attach&attach_id=11759 and https://web.archive.org/web/20161207060453/http://math.ucdenver.edu/~spayne/classnotes/09LinAlg.pdf .
Word of caution: As an algebraist, I found little of value in Axler's "LA done right". Your mileage may vary if you are chiefly preparing your students for (functional, in particular?) analysis.
