How should I approach the study of Linear Algebra? I'm a second-year college student having just previously finished calculus (derivatives and integrals, we never really got too far into multivariable calculus though we briefly went over some Taylor Series stuff) and I'm taking Linear Algebra right now.
I'm extremely confused, for a couple of reasons. 
The first reason I'm confused in the course is because I've never studied math correctly. I've done the bare minimum (memorize the steps to solve a problem type) to pass tests, I forget the steps right after the test, and I have never ever understood the reasons for the concepts I've learned in mathematics.
The second reason is because of the way I learn. I too often interpret concepts visually and desire concrete and literal visualizations, and I demand learning from the top-down. I've always been so frustrated that the way I've been taught - I argue the way most students have been taught - is from the bottom up: teaching one section at a time, with each concept seemingly isolated from each other, with each concept not fitting into a larger big picture, and then saying we "learned" . I'm not able to keep anything this way because I don't understand its context, its relation to other concepts that I've learned (that I will most likely soon forget anyhow), and the bigger picture in math of why this specific concept's invention was necessary. I can't tell you what a unit circle is for (other than passing corresponding set of problem types). I can look at a problem type and tell you it's familiar and look up how to "solve" this problem type, but I'm completely oblivious of any real mathematics.
I tried searching for a sane introductory description of Linear Algebra. All I can ascertain is that many problems in many fields of life and the sciences are discretely solved (broken-up, or simplified) by using linear algebra. But I don't really know what this means. Why was it invented? When I open up a book of Linear Algebra, I can't concretely/literally connect the first chapter on linear combinations (for example). Sure, if I looked a bit more, I could tell you how linear combinations might be connected to concept X... but X? How is concept X to linear algebra? And so on. These chain of relations form ... and I've never had a strong foundation to begin with. I can't digest or keep any information I've learned when I don't have a big picture overview of where to 'store' and 'process' the concepts I'm learning. Problem sets are specific to these mystery concepts, and again, I'm just memorizing steps to pass an exam. And then when I see fancy math language (proof language), it just makes me feel like I'm missing something obviously and grandly important.
Given these problems, could someone please explain how and why my approach is incorrect, where it's going to lead me (e.g. "I've seen many students take this path and they always end up..."), and how real mathematicians approach the study of not just Linear Algebra, but mathematics in general?
Edit: My major is Computer Science, and I am not looking to pursue mathematics professionally at the present time, but it seems awfully useful and I might want to minor in it.
 A: Here are some suggestions that may help.
$\bullet$ How to self study Linear Algebra. See my response and the other responses.
Study Habits from Math Stack Exchange
$\bullet$ What are some good math study habits?
$\bullet$ How to study math to really understand it and have a healthy lifestyle with free time?
$\bullet$ How can I learn to read maths at a University level?
Free Resources
$\bullet$ MIT Open Course Ware. You can also find others at other universities and these have free course notes and video lectures.
$\bullet$ Khan Academy. 
$\bullet$ Wiki Books Linear Algebra Resources
$\bullet$ Linear Algebra for Communications: A gentle introduction. I would study this to give you context!
$\bullet$ College Libraries: Peruse the books in the library and see if any fit the style you like.
Book Recommendations
$\bullet$ Linear Algebra Through Geometry, Thomas Banchoff, John Wermer (since you are visual)
$\bullet$ Practical Linear Algebra: A Geometry Toolbox, Gerald Farin, Dianne Hansford (since you are visual)
$\bullet$ Linear Algebra Done Right, Sheldon Axler
$\bullet$ Linear Algebra and Its Applications, David C. Lay
$\bullet$ Linear Algebra Textbook Recommendations on MSE
$\bullet$ Alternative to Axler's “Linear Algebra Done Right”
Problem Books
It is very helpful to do problems and practice, practice, practice to reinforce concepts!
$\bullet$ Linear Algebra Problem Book, Paul R. Halmos
$\bullet$ The Linear Algebra a Beginning Graduate Student Ought to Know, Jonathan Golan.
$\bullet$ 3,000 Solved Problems in Linear Algebra, Seymour Lipschutz
$\bullet$ Schaum's Outline of Linear Algebra Fourth Edition, Seymour Lipschutz, Marc Lipson
History of Linear Algebra
Linear Algebra History Websites
Regards
A: The problem with most text books is they are written not for students to learn from but for professors to impress each other with. This little gem was pointed out to me by my Antennas professor at university who's notes were spectacularly easy to follow and bore ZERO resemblance to the jargon filled ultra precise notation in Balanis.
Ok, so I'm nearing retirement and I never really took Linear Algebra because in engineering they "dog's breakfast" much of the material to shoe horn it into 4 years....so we had 2 weeks of linear algebra which was the normal pablem of determinants, Cramer's rule and multiplying matrices. But there is much more and I want to learn tensors so I need the "much more" bit.
So now I'm learning it and have found quite a few texts on line but the one I find the MOST useful is Ron Larson's "Elementary Linear Algebra". The second one I am using is David Lay's "Linear Algebra and Its Applications". Primary difference between Lay and Larson is the lay is much more proof oriented whereas Larson is not. I prefer Larson's layout and approach. Am also using Nicholson's "LINEAR ALGEBRA with Applications" which is available as a free download but he is very proof oriented and the layout is kind of like a handbook. I can see that Nicholson's book is a great reference if you already know the material but to learn from I found it wanting. The last book I was using was "Elementary Linear Algebra" By Author Wayne Roberts....1st edition. I have the students guide as well....they were have a book dump in the math dept at uni so I just took it before they threw it out. Roberts presents things very informally, I can see where some people would like that but I don't.
So what I do it take notes from Larson, do all the problems. Then do the problems from Lay, then try and don the problems from Nicholson. I a bit dumb so I need to do lots of problems.
The great challenge with linear algebra is that its tedious which makes it difficult...at least that was the case for me. Differential equations and calculus were different.
Anyhow, hope this info helps.
Cheers,
Jim
