I've taught Linear Algebra a number of times. Teaching students to deal with proofs is an important part of that course. Luckily the textbook (Linear Algebra and its Applications 4th Edition, by David C. Lay) had true false questions in every section, and a bunch more at the end of each chapter. I used those to help students think carefully about the ...
Here are some ideas:
As Brendan says above, you can give an example or non-example, and ask the student to explain why (using the definition) it is an example or non-example.
You can ask for examples satisfying certain properties ("Give an example of a relation on a three element set which is transitive and symmetric, but not reflexive").
You can ...
I'll reiterate my recommendation of MyOpenMath. To your points:
It understands TeX (a simplified version, using ` math ` instead of \$ math \$)
It sllows the use of parameters in some way to randomize numbers for each student
It allows these parameters within the TeX code.
It can be integrated into Blackboard (that's how I've been using it) using an LTI ...