"almost certainly" may be a bit strong. Here's what I found when looking at a few applied math programs wrt algebra (not linear)
Required
MIT
Berkeley
Not Required
UCLA
Washington
Caltech
NYU
Harvard (SEAS)
Princeton
Brown
Duke
I know someone from UCLA that got into a top 10 applied math grad program without taking algebra (or any of 134-136 either), so it's certainly possible. I would also suggest taking the honors sequences that are available. I don't think the order is a big deal for most of those classes.
You can see actual class sizes for the math department here: http://www.registrar.ucla.edu/schedule/detmain.aspx?termsel=11F&subareasel=MATH , http://www.registrar.ucla.edu/schedule/detmain.aspx?termsel=12w&subareasel=MATH ...
I took a 'Careers in Statistics' seminar course, and almost all of the presenters said you need more than Bachelor's to do "interesting" statistical work. That may have just been a matter of bias (most of them had PhD's, all had some advanced degree) - there certainly seem to be good job...
Have you looked at Enderton's book on Mathematical Logic? I haven't read that one myself, but it seems to be widely used, and I really liked his book on Computability Theory.
https://www.amazon.com/dp/0387708723/?tag=pfamazon01-20
This book is geared toward statistics, but it seems to be pretty good for most applied purposes. The order in which topics are presented is a bit idiosyncratic, so it may not be a good choice for supplementing a class that uses a...
At my school, the lower division class is basically matrix algebra. The upper division class is basically the same, except in the context of vector spaces and with all results proven. The upper division class is meant to be an introduction to proofs as well.
Every math program whose requirements I've specifically looked at require a foreign language or two. Usually they allow French, Russian, German and a few other human languages. Never something like C++. Often the form of the requirement just seems to be transcribing a research paper that was...
(I'm putting this here as opposed to the Career forum because it's for REU applications)
I have some experience in C++, MATLAB and R, but I'm not sure how to indicate on the CV how much experience. I'm not an expert in any of them, but I don't know that "beginner" is right either. Years...