Page 101 - 35Linear Algebra

5

Vector Spaces

n
As suggested at the end of chapter 4, the vector spaces R are not the only
n
vector spaces. We now give a general definition that includes R for all
S
values of n, and R for all sets S, and more. This mathematical structure is
applicable to a wide range of real-world problems and allows for tremendous
economy of thought; the idea of a basis for a vector space will drive home
the main idea of vector spaces; they are sets with very simple structure.
The two key properties of vectors are that they can be added together
and multiplied by scalars. Thus, before giving a rigorous definition of vector
spaces, we restate the main idea.

A vector space is a set that is closed under addition and
scalar multiplication.

Definition A vector space (V, +, . , R) is a set V with two operations +
and · satisfying the following properties for all u, v ∈ V and c, d ∈ R:

(+i) (Additive Closure) u + v ∈ V . Adding two vectors gives a vector.

(+ii) (Additive Commutativity) u + v = v + u. Order of addition does not
matter.

(+iii) (Additive Associativity) (u + v) + w = u + (v + w). Order of adding
many vectors does not matter.

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