Vector Spaces

Vector Space

Let be a field. A -vector space is a set with two operations:

• Addition:
• Scalar multiplication:

which satisfy the following properties:

• Associativity:
• Commutativity:
• Additive Identity:
• Additive Inverse:
• Multiplicative Identity:
• Distributivity:

e.g. The set is a vector space on under the addition and scalar multiplication defined by $$(f+g)(x)=f(x)+g(x),\quad (\alpha f)(x) = \alpha f(x)$$$C[a,b]={f\colon[a,b]\to\R\mid f \text{ is continuous}}$ is a subspace under the same addition and scalar multiplication.

Proposition

Let be an -vector space, , then

Proof . It follows that $$0=0_{F}\cdot v+ (-0_{F}\cdot v)=0_{F}\cdot v$$$\square$

Subspace

A subspace of a vector space is a subset which is also a vector space inherit the same operations as in .

Proposition

A subset is a subspace of a -vector space if and only if

Span and Linear Combinations

Span

The span of a set of vectors is the set of all finite linear combinations of vectors in :

Proposition

The span of any nonempty set of vectors is a subspace.