Math Teachers: How much theory do you cover in your Linear Algebra?

As someone who has taught this class in the past, it's difficult to put too much theory at the students. Nonetheless, I expose the students to the vector space axioms and emphasize that there are other examples of vectors out there besides the "arrows" in R^n.

I still have them do basic  basis, vector subspace, and linear transformation proofs. I do sneak at least one polynomial vector space problem in there from time to time to break up the R^n "monotony". Finally, I have the students do basic eigenvalue/ eigenvector proofs (the computational ones that can be done in a few lines). It's very difficult to have them do purely logical paragraph proofs at this early stage; so I stick to more "computational" proofs as a way to transition them into this kind of thinking.

I hope that addresses your questions. Have fun!

I hardly paid attention to my Algebra I class two years ago, my parents taught me how to do linear algebra.  Simply tell them about rise over run, how lines go up when they're positive and down when negative.  How to interpret a linear equation in all their forms.  Also those mathemaical proofs you're talking about are pretty much logic and order of operations imo.  They're going to hate you later if you call those proofs now, what will  they call proofs in Geometry D:  ALgebra is a piece of cake, just make sure they learn all the correct ways to map a graph and solve an equation and they'll be fine.  My teacher used to make cheezy jokes about graphing as well, it helps sometimes definitely when you start graphing more than just parabolas and hyperbolas.  What do you mean theory and applications... Teach them how all of it works by giving them problems to solve, that's the best way to prepare people.  Also most text books that schools use should be sufficient, I find Holt to be pretty good.  Borrowing a random book from the library shouldn't do much harm though, most text books are all the same.

Exactly, many students hate proofs,i usually gave 60% theory