What is the most tangible way to introduce linear transformations in a linear algebra course? Most books tend to take a very abstract approach to this topic.
I think linear transformations are abstract by nature. Of course you can construct geometrical analogies in many cases, like for projection operators, rotations etc, and you might be able to use such examples to guide students towards the general definition. Remember the "mathematical programme": Ideas -> constructions -> abstraction -> special cases :)
i used to try all kinds of examples of linear phenomena. E.g. cooking recipes. Doubling the ingredients of the recipe doubles the output. the main job is to convey the idea of linearity, outputs that change proportionately to the inputs. then a linear transformation is any operation that behaves like this. e.g. differentiation behaves linearly on functions. projections of one space onto a lower dimensional space are linear. but force is not linear with speed, i.e. F = MA, so force is proportional to acceleration.
I'd introduce them by showing what they actually do, take you from one space to another. The way Gilbert Strang does it on the MIT opencourseware linear algebra course is pretty good if you want to get introduced to what they do imo
Since a primary application is to differential equations, with students who have had calculus it seems important to point out that differentiation is linear. When acting on polynomials of fixed degree it also gives the basic example of a nilpotent linear operator, not an intuitive idea without that example. And when acting on spaces of exponential functions it gives the fundamental example of eigenvectors and eigenvalues, another absolutely crucial concept to acquire.
I learned linear algebra best when I thought in terms of geometry. Unfortunately, linear algebra starts in R^{n} from the start which is pretty annoying from someone like me. I made everything into a simpler case in R^{2} or R^{3}. Without writing my own thoughts I found a good link for how I would best learn this. http://www.math.hmc.edu/calculus/tutorials/lineartransformations/ If this is your first exposure to linear algebra I would highly recommend this book: http://www.amazon.com/Linear-Algebra-Modern-Introduction-CD-ROM/dp/0534998453 The price is slowly going up because the editions are getting farther along. I have the 2nd edition and it's wonderful for showing the intuitive and visual representation of linear algebra. This is how math should be taught.. at least for learners like me.