Linear Algebra: Quotient Spaces

Equivalence Relation in Vector Spaces

• Define equivalence relation on vector space ( V ): ( v_1 \sim v_2 ) if ( v_1 - v_2 \in W ), where ( W ) is a subspace of ( V ).

Quotient of Vector Space

• The set of equivalence classes ( V/\sim ) is the quotient of ( V ) by its subspace ( W ), denoted ( V/W ).

Vector Space Structure of Quotient

• Quotient ( V/W ) forms a vector space.
• Define equivalence class of ( v \in V ) as ( v + W ).
• Vector addition: ( (v + W) + (v' + W) := (v + v') + W ).
• Scalar multiplication: ( a(v + W) := av + W ).
• Well-definedness confirmed by the property of equivalence relations.

Linear Map to Quotient Space

• A linear map ( V \to V/W ) exists, mapping each vector ( v ) in ( V ) to its equivalence class ( v + W ).
• The kernel of this linear map is the subspace ( W ).

Isomorphism Between Image and Quotient

• The image ( \text{Im} f ) of a linear map ( f: V \to W ) is isomorphic to ( V/\text{Ker} f ).

Proof of Isomorphism

• Define map ( f: V/\text{Ker} f \to \text{Im} f ) by ( f(v + \text{Ker} f) = f(v) ).
• Check injectivity: If ( f(v + \text{Ker} f) = 0 ), then ( v \in \text{Ker} f ).
• Check surjectivity: For any ( w \in \text{Im} f ), there exists ( v \in V ) such that ( f(v) = w ).

Dimension Formula

• For a linear map ( V \to V/W ), it is surjective with kernel ( W ).
• Dimension formula: ( \text{dim}(V/W) = \text{dim}(V) - \text{dim}(W) ).

Example of Dimension Calculation

• For space of polynomials degree ( \le n ), ( \text{dim}(V) = n + 1 ).
• If linear operator ( f: p(x) \to p''(x) ), kernel is the subspace of polynomials degree ( \le 1 ) (dim ( 2 )).
• Thus, ( \text{dim}(V/W) = n - 1 ).

Well-defined Operator Induction

• A linear operator ( f: V \to V ) induces a well-defined operator ( \overline{f}: V/W \to V/W ) if ( W ) is ( f )-invariant.

Requirements for Well-definedness of Induced Operator

• The function ( \overline{f} ) is well-defined if for all ( v + W = v' + W ), ( \overline{f}(v) + W = \overline{f}(v') + W ).
• This requires ( f(v) - f(v') \in W ).

Basis and Induced Operator

• For finite-dimensional ( V ) and ( f )-invariant subspace ( W ), if ( B_W ) is an ordered basis for ( W ):
  • Extend ( B_W ) to an ordered basis ( B ) for ( V ).
  • ( A = {v + W : v \in B \setminus B_W} ) forms an ordered basis for ( V/W ).
  • Relation between matrices: ( M_{B}(f) = M_{B_W}(f) ) follows from the induced operator on ( V/W ).