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Algebra can be used in decision making, data entry, calculating distance, and many other areas.
When focusing upon abstract algebra, there are many different areas included within this topic such as groups, rings, modules and vector space. These all are part of the sequence to constructing abstract algebra.
Linear transformations can be very important in graphics. Also, linear transformations come up whenever you need to solve systems of linear equations, which arise quite often. Finally, they can be useful in further areas of mathematics such as topology.
Yes. I use them a lot when answering these questions.
Algebras are systems of rules for manipulating strings of symbols. The most familiar algebra taught in school is used for manipulating ordinary mathematical equations for calculating numeric quantities, but there are many other algebras (e.g. boolean algebra, knot algebra, text parsing algebras, hoph algebras, quaternion algebra, group algebra, cellular algebra).