Abstract Algebra/Vector Spaces

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[edit] Definition

Definition (Vector Space)
Let F be a field. A set V with two binary operations: + (addition) and \times (scalar multiplication), is called a Vector Space if it has the following properties:
  1. (V, + ) forms an abelian group
  2. v(a + b) = va + vb for v \in V and a, b \in F
  3. a(v + u) = av + au for v,u \in V and a \in F
  4. (ab)v = a(bv)
  5. 1Fv = v

The scalar multiplication is formerly defined by F \times V \xrightarrow{\phi} V, where \phi((f,v)) = fv \in V.

Elements in F are called scalars, while elements in V are called vectors.

Some Properties of Vector Spaces
  1. 0Fv = 0V = a0V
  2. ( − 1F)v = − v
  3. av = 0 \iff a = 0 or v = 0
Proofs:
  1. 0_F v = 0_F v + 0_V = (0_F + 0_F) v = 0_F v + 0_F v \Rightarrow 0_V = 0_F v . Also, a 0_V = a 0_V + 0_V = a(0_V + 0_V) = a0_V v + a0_V \Rightarrow a0_V = 0_V v
  2. We want to show that v + ( − 1F)v = 0, but v + ( − 1F)v = 1Fv + ( − 1F)v = (1F + ( − 1F))v = 0Fv = 0V
  3. Suppose av = 0 such that a \neq 0, then a^{-1} (a v) = a^{-1} 0 = 0 \Rightarrow 1_Fv = v = 0
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