Subspaces and the basis

Subspaces

• \(S\) is a subset of \(V\).
  1. S ⊆ V
  2. S is a vector space
     1. include zero vector
     2. closed under addition
     3. closed under scalar multiplication

Basis

• minimum set of vectors that spans the subset
• \(S\) is a basis of \(V\) ⟺
  1. elements of \(S\) are linearly independent
  2. \(S\) spans \(V\)
• 특정 부분집합의 basis의 linear combination으로 표현되는 모든 벡터는 유일하다.