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A foundational textbook on abstract algebra with emphasis on linear algebra.

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Linear Algebra by Jim Hefferon

Elementary Linear Algebra by Keith Matthews

Linear Algebra, Infinite dimensions and Maple by James Herod

Elementary Number Theory by William Stein

Abstract Algebra: The Basic Graduate Year, A Course in Algebraic Number Theory, and, A Course in Commutative Algebra are three e-books by Robert Ash and are available here on his website.

A Course in Universal Algebra by Stanley Burris & H. P. Sankappanvar

Linear Algebra by Leif Mejlbro

Leif has published many of his books, sixty-six to be precise, freely on Bookboon.com . You can see what he offers freely at http://bookboon.com/en/search?q=author%3A%22Leif%20Mejlbro%22 . In his Linear Algebra book-series he offers great details and excellent write up. He has aimed this series to be a practical guide for students in Physics and the technical sciences. For that reason the emphasis has been laid on worked examples, while the mathematical theory is only briefly sketched without proofs.  There are total three books on Linear Algebra, all hosted at Bookboon.com all of which can be downloaded freely.

• Linear algebra c-1 – Linear equations, matrices and determinants
• Linear algebra c-2 – Geometrical Vectors, Vector Spaces and Linear Maps
• Linear algebra c-3 – The Eigenvalue Problem and Euclideam* Vector Space

Four other books on Linear Algebra by Kenneth Kuttler are also considerable.

Group theory for Maths, Physics and Chemistry by Arjeh Cohen, Rosane Ushirobira, Jan Draisma

Symmetry plays an important role in chemistry and physics, both at the macroscopic and the microscopic level. Group theory is an abstract setting capturing the symmetry in a very efficient manner, which helps to make computations more efficient. We focus on abstract group theory, deal with representations of groups by means of permutations and by means of matrices, and deal with some applications in chemistry and physics.
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Sets, Relations, Functions by Ivo Duentsch and Günther Gedigo
Group Theory by Pedrag Civitanovic

Linear & Multilinear Algebra by C C Wang & R M Bowen

Abelian Categories by Peter Freyd

Categories and Groupoids by P. J. Higgins

Lie Algebras by Prof. Sternberg

Intro to Abstract Algebra by Paul Garrett

The text covers basic algebra of polynomials, induction and the well-ordering principle, sets, counting principles, integers, unique factorization into primes, prime numbers, Sun Ze’s theorem, hood algorithm for exponentiation, Fermat’s little theorem, Euler’s theorem, public-key ciphers, pseudoprimes and primality tests, vectors and matrices, motions in two and three dimensions, permutations and symmetric groups, rings and fields, etc.
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Algebraic Topology by Allen Hatcher
A Topology Primer by Klaus Wirthmüller

Infinite-dimensional Lie Algebras by Iain Gordon

Notes on the course Algebraic Topology by Hal Sadofsky

This list is expandable. If you know any other book on Algebra which is available online for free, then please give a few seconds and put that into the Comment-Box below, with the link. (It supports basic HTML and Markdown writing.)