I challenge the assertion that students need to see applications in everything.
When I first started teaching I labored under the delusion that I should explain connections to physics whenever I could (in calculus, DEQns, linear algebra etc.). Now, I think my efforts do have an audience, but not the main audience that I find in my classes. Personally, I've always had a fairly big majority of engineering students with a sprinkle of math, math-education, comp-sci, various science majors. It turns out, the fear of math is only outdone by the fear of physics. I've learned to go easy on those comments in recent years. If I was to teach an audience of math-physics undergraduates then you would hardly recognize the course in comparison to what I teach now.
My approach is simple. In math class we talk about math.
I will make time for applications in maybe 10% of the time, but I refuse to let it take too much time. If you pick a particular application then that audience will love it, but the rest of the kids are then bored, frightened or annoyed. The common thing all students have in taking a general education math course is the need to study math.
To my thinking, homework is the place for applications. How can they use math from class to solve those beloved real world problems? If this homework is recommended and not required then the audiences which express this supposed love of applications can pursue their passion individually.
There is plenty of theory in linear algebra to fill the semester. I think by in large homework is the place to play with applications. For example:
- in class I teach the method of row reduction
- in homework I ask them to find polynomials to fit data
Now, I will talk applications in passing, but not as motivation. Motivation is by in large given from mathematical questions. It might be born of an example, but not an application. Motivating a general mathematical topic from a particular application can give a very skewed origin story.
All of this said, if I was teaching "Applied Linear Algebra" then my focus is quite different. So, the question is, what is the course description?
Anyway, the assertion that math should be taught always without applications is just as absurd as the incessant drumbeat to include applications at the cost of technique and analysis. Balance, moderation, academic freedom for a better future.
EDIT: after reading some answers I noticed I missed a major point in responding to your question. By your account, you were in the middle of Lecture stating the central core definition of the course when you were interrupted with this purportedly genuine question from the student. It occurs to me it doesn't actually matter if the question is sincere or not. The fact is that this student has violated a basic part of the student/professor relationship. Asking you to change the direction of a lecture mid-stream because he is not particularly excited about the core of a class? What if I did the same in an applications class? How about in the middle of my physics professor working an example I interupt and ask why we are not focusing more on theorems of physics which apply to all problems as opposed to this particular problem which has no bearing on the larger picture? Is it ok for me to demand more theory in the midst of a carefully prepared difficult example the professor has labored to prepare? Of course not. That would show a fundamental disrespect for the professor. I'm not much for holding up professors in terms of academic rank, but for the student/professor relationship to work it is also necessary for students to consider the humanity of professors. In short, I think the best response to such a student when the question happens is: "you are welcome to visit office hours, I'd be happy to discuss applications with you there. Our focus here is theory and in time I'll show you how this theory allows endless application to applications..." something like that.