Semester 4 / Linear Algebra / Vector Space

Vector Space

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(V,∗,∘)  over  (F,+,⋅)(V,*,\circ)\;\text{over}\;(F,+,\cdot) is a vector space, iff it satisfies:

  • (V,∗)(V,*) is an abelian group
  • (F,+,⋅)(F,+,\cdot) is a field
  • ∀a∈F,∀x∈V,a∘x∈V\forall a \in F, \forall x \in V, a \circ x \in V
  • ∀a∈F,∀x,y∈V,a∘(x∗y)=(a∘x)∗(a∘y)\forall a \in F, \forall x,y \in V, a \circ (x * y) = (a \circ x) * (a \circ y)
  • ∀a,b∈F,∀x∈V,(a+b)∘x=(a∘x)∗(b∘x)\forall a,b \in F, \forall x \in V, (a+b) \circ x = (a \circ x) * (b \circ x)
  • ∀a,b∈F,∀x∈V,(a⋅b)∘x=a∘(b∘x)\forall a,b \in F, \forall x \in V, (a\cdot b) \circ x = a \circ (b \circ x)

In the above definition:

  • FF is the set of scalars
  • VV is the set of vectors
  • ++ is number addition
  • ⋅\cdot is number multiplication
  • ∗:V×V→V*: V \times V \to V is vector addition
  • ∘:F×V→V\circ: F \times V \to V is scalar multiplication

Alternate Notation

When no confusion arises, instead of 4 different symbols:

  • ++ for both vector addition and scalar addition
  • ⋅\cdot for both scalar multiplication and field multiplication

And:

  • Zero vector is denoted as 0‾\underline{0}
  • (V,∗,∘)  over  (F,+,⋅)(V,*,\circ)\;\text{over}\;(F,+,\cdot) is simply denoted as VV over FF.

Properties

Suppose (V,∗,∘)  over  (F,+,⋅)(V,*,\circ)\;\text{over}\;(F,+,\cdot) is a vector space.

Zero Vector Exists

Denoted as ee.

∃e∈V such that ∀x∈V,  x∘0=e\exists e \in V \text{ such that } \forall x \in V,\; x \circ 0 = e ∀a∈F,  a∘e=e\forall a \in F,\; a \circ e = e ∀a∈F,∀x∈V;  a∘x=e  ⟹  a=0  or  x=e\forall a \in F, \forall x \in V;\; a \circ x = e \implies a = 0\;\text{or}\;x=e

Negative x Is the Additive Inverse

x‾=(−1)∘x\overline{x} = (-1) \circ x
Written by September 29, 2026 2 min read
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