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I have finished my universities Linear Algebra 1 but I do not feel satisfied with the understanding I left with ( the class was a bit "light" but also me neglecting the need for a true understanding didn't help ). I would like to work through Linear Algebra Done Right by Sheldon Axler also.

I need help developing my skills at constructing proofs before I start Spivak's book. Before I do that though I would like to get some other books under my belt.

Right now I am looking at:

A Concise Introduction to Pure Mathematics - Martin Liebeck

http://www.amazon.com/dp/1584881933/?tag=pfamazon01-20

It is meets all my criteria for what I want except there is no solution manual to be found, this is a problem as I am going to be self-studying. I have only read a bit of it and it seems alright, besides the constant uncertainty when I solve a problem.

Another book I have been looking at is

Chapter Zero : Fundamental Notions of Abstract Mathematics - Carol Schumacher

http://www.amazon.com/dp/0201437244/?tag=pfamazon01-20

It seems like a bit much for me to jump into though I think I could handle it if I gave it some time. A instructor at my university has a set of lecture notes that "sort of" accompany it, so that might help. I can not seem to find a solution manual to this also, another problem.

A book on Proofs I have been looking at is :

The Nuts and Bolts of Proofs - Antonella Cupillari

http://www.amazon.com/dp/0120885093/?tag=pfamazon01-20

This book seems pretty good but I would like to do/read a bit more set theory and logic before starting it. Although it does seem manageable. It has detailed solutions in the book itself, which is a large plus.

This also seems to be along the lines of what I am looking for :

How to Prove It: A Structured Approach

http://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

I have yet to actually hold the book and look through it, I am also unsure if there are solutions out there for it.

I would really appreciate suggestions on books that encompass my current interests anything that will help me get started with pure mathematics. Specifically develop my skills at writing proofs, and obviously some of the prerequisite fundamentals needed to write them.

My current pre-occupation with desiring ( needing?) a solution manual for "security" may be holding me back, but any texts you could recommend would be greatly appreciated.