Linear Algebra: Quotient Spaces

Study Notes

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 ).

Description

Explore the concepts of quotient spaces in linear algebra through this quiz. Learn about equivalence relations, subspaces, and the properties of vector spaces. Test your knowledge on defining and proving key aspects of quotient structures.