STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

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2. Vector spaces
• Now let’s relate matrices to vector spaces. We
start with the definition of a vector space. Thisis
largely for formal completeness - you might wish
to skip over the next bullet - since the only vector
space considered here will be
R = all −dimensional vectors with real elements
and its subspaces.
• We list a number of axioms to be satisfied by a
structure in order that it be called a vector space;
for R these are all pretty obvious. Note that
R is closed under addition (x y ∈ R ⇒ x +
y ∈ R ) and scalar multiplication (x ∈ R and
∈ R ⇒ x ∈ R ), and satisfies
1. Associativity: For all x y z ∈ R , we have
x +(y + z) =(x + y)+z.
2. Commutativity: For all x y ∈ R ,wehave
x + y = y + x.