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"Algebra" by Michael Artin is a classic. "Abstract Algebra" by Dummit and Foote is also great. Technically D&F is a graduate level book, but that's just because it contains so much content (it's over 1000 pages!). One can easily learn undergraduate level algebra from it by only studying the appropriate sections.

For linear algebra I recommend Axler's Linear Algebra Done Right.

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Q: What is the best college algebra book?
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