1. The document defines a vector space as a nonempty set of vectors where two operations, vector addition and scalar multiplication, satisfy ten axioms.
2. A subspace is defined as a subset of a vector space that contains the zero vector and is closed under vector addition and scalar multiplication.
3. The span of a set of vectors is the set of all possible linear combinations of those vectors, and the span of any set of vectors in a vector space forms a subspace.
4. VECTOR SPACES AND SUBSPACES
• Definition: A vector space is a nonempty set V of
objects, called vectors, on which are defined two
operations, called addition and multiplication by
scalars (real numbers), subject to the ten axioms
(or rules) listed below. The axioms must hold for all
vectors u, v, and w in V and for all scalars c and d.
1. The sum of u and v, denoted by is in V.
2. .
3. .
4.
.
u v
u v v u
(u v) w u (v w)
u ( u) 0