# Proofs in Linear algebra

1. Oct 20, 2015

### MostlyHarmless

I'm grading for a linear algebra class this semester. The class is comprised entirely of engineering majors of various flavors. The hw assigned by the professor is almost entirely "proofs" they are fairly specific proofs. Really the only thing that designates them as proofs is that the questions start with "show" instead of "find". I feel like this is largely unfair to the students. As I'm told to not only grade their answers, but their proofs, but these are engineers.. They've never taken any proof based classes and never will after this. In fact, the math majors are required to take an alternative, proof based linear algebra class.

Also, they are horrid at proving things. Which, why shouldn't they be? They are engineers, not math majors. So it makes everyone's life harder than it needs to be.. In my opinion at least.

I'm interested to hear what others think about making engineers prove things.

2. Oct 20, 2015

### Simon Bridge

It's a good idea - and quite normal in NZ colleges.
Did you check what counts as a proof for the class in question? i.e. did you review the model answers and/or the course material?

3. Oct 20, 2015

### MostlyHarmless

I'm given solutions for each homework. These solutions are generally less "rigorous" than a proof that I would turn in on any of my own homeworks. So I don't expect the proofs on the homeworks I'm grading to be something like: "Let blah be a blah such that blah.. then blah, therefor blah, and hence blah.." I generally just look for a solid progression of justification with some words, or no words if the symbol pushing is clear enough.

My main complaint though, is it seems like there is nothing being said in the class about writing proofs. I make small notes when I can, but I continually see instances where the student likely understands, but they omit a critical step of a proof because it is "obvious". So that effectively miss the entire point of the proof. For example: In showing that addition is commutative in $\mathbb{C}$. They will do all the associating of of the reals and imaginary parts, but then they do this $(a+c) +(b+d)i = (c+di)+(a+bi)$. Which is obvious, but the point is that complex addition commutes because real addition does. Which is lost here.

4. Oct 21, 2015

### Simon Bridge

It would appear, therefore, that a rigorous or formal proof is not required for the class.
You are over-thinking the assignment - not everyone uses exact language all the time.

5. Oct 21, 2015

### MostlyHarmless

I most certainly am over-thinking the assignment, that was never really in question. My point is, it seems like it would be more helpful to ask them to just calculate things than it is to ask the students to prove things, but then allow for a proof that skims the ideas without forcing them to explore exactly why something is true.

6. Oct 21, 2015

### Staff: Mentor

IMO, it's a good idea. At some point in their (engineers) career, they will likely need to justify (i.e., "show") why their new idea is reasonable and valid.

It's not just engineering students who have a hard time with linear algebra with its problems requiring proofs -- math students typically have difficulties as well, in my experience. What you might do is to talk with the instructor for the course, about making available a resource on some basic types of proofs, with a few examples of each type.

7. Oct 22, 2015

Staff Emeritus
I don't think "show that" is really identical to "prove". I think "show that" allows for a certain lack of rigor, or as I would rather put it, reliance on a past body of work. I had a rather "proofy" linear algebra class, and I thought it was valuable - but it focused on the main ideas, and not the minutiae that can easily crop up in a proof-based class.

8. Oct 22, 2015

### vela

Staff Emeritus
I agree. Critical thinking is a skill engineers should be developing. Being able to write a proof may not be a skill they need, but they should know enough to be able to justify what's "obvious," rather than just accepting something on faith. Writing valid proofs will help them learn how to do that.

When I was an undergrad, I took a course in set theory, which was the course most of the computer science majors took to satisfy the requirement of taking an upper-division math course. The professor gave weekly quizzes where students were asked to write proofs. The results of the first quiz, in his words, was "a disaster." He said 3 students out of about 40 did well on the quiz, and they didn't need to stick around for class that day. Everyone else failed, and he was going to teach them how to write a proper proof. After the second quiz, he said people did much better.