Restart mathematics
Solution 1:
First I'm impressed with your strong desire to learn linear algebra and refresh your mind, good luck Sir!
For the book recommendation, I highly recommend this book, which I used during my learning trip through this important topic: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares this is the link where you can buy it.
I wish that you'll find all your questions answered there, and again good luck!
Solution 2:
You used the "linear algebra" tag for your question, but in your question you asked about algebra. They are different subjects; linear algebra is more advanced. I am going to assume that you are interested in basic algebra.
I have no idea if it's the best book, but I've found Basic Mathematics by Lang to be a reasonable introduction.