Originally Posted by mantrapad
I've found this regular definition of vectors: A quantity which has both magnitude and direction.
On some sites, I've also found the addition: They satisfy the vector laws of addition (commutative, associative, distributive).
Is it really necessary that all vectors should satisfy all those laws? Are there any exceptions?
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The "magnitude and direction" definition is an intuitive one. They make a lot of sense for R^n, but for some vector spaces, it doesn't. There are function vector spaces, where we create rules for adding and scaling functions. For example, if f(x) = x^2 and g(x) = x + 1, then (f + g)(x) = x^2 + x + 1. But what "direction" does f, g, or f+g point? It's much more abstract!
