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I sometimes feel that proofs crowd out a coherent vision for linear algebra. However I also think a central theme of a Linear Algebra course is to learn reasoning even though it does not always succeed.
The audience is first year undergraduate students studying mathematics and physics but maybe extended to engineers. They generally struggle with the idea of proof.
The question can be modified: what is the purpose of teaching any proof at all, in any mathematical course?
The answer of course is heavily dependent on the institute, the crowd, and the point behind the course.
My experience is that non-mathematics students often see mathematics as a tool and that's it. In their minds you just need to know some basic facts and then use that for the sake of engineering or physics or other mission to accomplish in a mathematical fashion.
Those students will mostly misunderstand proofs, misunderstand definitions, and will generally be unable to see the full depth of theorems (due to lack of proper definitions, due to the way they study, or due to the fact they simply don't care). However some of the students will be very receptive and will understand the proofs and their inherent beauty.
In those courses one can say that there is little to no point in teaching proofs. However the idea is that you teach reasoning, and you give these proofs as an example of a proper mathematical reasoning. This reasoning is very important because it allows you later on to examine things others will tell you.
Of course, if a student cares little about this reasoning and only wants to learn the names for the black boxes which solve problems - it will not stick.
On the other hand, if the course is for mathematics undergrad students then they have to see the proofs and they have to learn the reasoning. Often, too, they will have other courses in which proofs are presented and reasoning is discussed and this will help to engrave these processes deeper into their minds.
Another very important reason to teach proofs is to get students used to the fact that in mathematics you don't usually rely on others in this aspect, you have to understand the proof given to you in order to truly understand something. You don't accept things, you find out why they are true on your own.
For these reasons in my university engineering students take only one course in linear algebra but mathematics students have to take two.
In my first year engineering math we did the $\epsilon-\delta$ definition of a limit, but only in multivariable calculus. We went over the proper definition of the Riemann integral as well. Yeah, it was weird.
The idea seems to be that proofs are presented when they fit nicely into an understanding of how something works, but if the concept can be intuitively worked with and 'understood' without proof (like in the single variable case), they don't bother. Our linear algebra exposure was just some matrix mechanics, and mushed into the second half of our multivariable calculus course. Not well planned.
So I would use that as a guiding principle. How much do these proofs help you understand how to use the concepts, and how much are they simply for rigor? I think that proofs can have a place in such a course, but that their use should be more carefully justified when the course isn't for math students.
It's useless to explain the proof of a theorem to engineers, who hardly know how to define $\mathbb{N}$ - or do any axiomatic mathematics at all, for that matter. Either give examples to show that it's true, or start from scratch entirely (which I suppose you do not have time for).
Mathematicians obviously need to see the truth.
What physicists need to see depends on what kind of physicist they want to be. If they'll become string theorists, a strong mathematical background is useful, but you don't need to know any specific proofs of theorems for applied classical mechanics.
I see what you mean sometimes linear algebra can seem too proof heavy in that it makes approaches to pragmatic problems crowded by proof like thinking when all you want is an algorithm or process to get the required result. However one of the major themes of modern mathematics is the classification of structures and objects and using the proofs from linear algebra is an important tool to tell us when two objects are apparently different objects or are indeed the same.
Proofs will let you find out if the students understand mathematics on a deeper (or higher?) level. I think this insight is valuable in any student and that proofs need a some emphasis in any linear algebra course. I don't think I have to tell anybody that this insight is more vital in mathematics students than in others, and that therefore proofs need have more weight in their grading. I just mention this in order to make myself clearer to the downvoter.
I would say that all engineering students here, have the background for understanding mathematical proofs.
Really? In my experience, many students have trouble with induction, so you will forgive me for not quite buying that first-year undergraduate students at your institution have the background for understanding mathematical proofs.
I think an introductory linear algebra course doesn't need much in the way of proofs - the basic stuff is really quite easy to prove.
In my opinion, a course specifically targeting proofs should be offered if the department considers proofs to be important enough (and they certainly are, especially if you're offering a degree in mathematics). Now, what constitutes a decent proof for such a course is up to debate.
