Questions and Answers
What is the equivalence relation defined on V?
v1 ∼ v2 if v1 − v2 ∈ W
What do we call the set of equivalence classes V/∼?
The quotient of V by its subspace W, denoted by V/W
How is the equivalence class containing v ∈ V denoted?
v + W
What linear map exists from V to V/W?
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What is the relationship between Imf and V/Kerf?
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What must be checked to prove that Imf is isomorphic to V/Kerf?
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What is the kernel of the linear map from V to V/W?
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What is the dimension of V/W in relation to V and W?
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When does a linear operator f induce a well-defined linear operator on V/W?
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What is required for the function f̄ to be well-defined?
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What kind of basis does the set A = {v + W : v ∈ B \ BW} form for V/W?
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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 ).
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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.