Quotient Space Linear Operator

In summary, the conversation is about solving a problem involving the operation (A + B)/B → A/(A \cap B), where A, B are subspaces of V and a, b are elements of A, B respectively. The operation is shown to be linear and bijective, and it is explained that F(v) = a when v is written as a+b. The concept of quotient space is mentioned and the speaker thanks the other person for their help.
  • #1
schw
3
0
Could anyone help me solve this problem?

Let A,B be two subspace of V, a \in A, b \in B. Show that the following operation is linear and bijective:

[itex](A + B)/B → A/(A \cap B): a + b + B → a + A \cap B[/itex]

I really couldn't understand how the oparation itself works, i.e, what F(v) really is in this problem.
 
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  • #2
write v as a+b, then F(v) = a. now chck this is well defined, i.e. if a+b ≈ a'+b' modb, then a ≈ a' mod (AmeetB).
 
  • #3
I just have to ignore the sets that are in the operation? (v = a + b + B)

edited:
Just realized the definition of quocient space. Tahnks for helping
 
Last edited:

1. What is a quotient space linear operator?

A quotient space linear operator is a mathematical concept used in linear algebra. It represents a mapping between two vector spaces where the output space is a quotient space of the input space. This operator is used to define a relation between vectors in the input space that are identified as equivalent in the quotient space.

2. How is a quotient space linear operator represented?

A quotient space linear operator is typically represented by a matrix. The rows of the matrix correspond to the basis vectors of the quotient space, and the columns correspond to the basis vectors of the input space. The entries of the matrix represent the coefficients of the linear combination of the input space vectors that produce the output space vector.

3. What are the applications of quotient space linear operators?

Quotient space linear operators have numerous applications in mathematics and physics. They are widely used in differential geometry to study manifolds and in functional analysis to study Banach spaces. They also have applications in quantum mechanics, where they are used to describe the symmetries and transformations of physical systems.

4. How is a quotient space linear operator related to a quotient space?

A quotient space linear operator is defined on a quotient space, which is a mathematical concept that represents the space of equivalence classes of a given vector space. The operator maps vectors in the input space to equivalence classes in the quotient space, and vice versa. This relationship allows for the study of the properties of the quotient space through the use of linear operators.

5. What are the properties of a quotient space linear operator?

A quotient space linear operator has several important properties, including linearity, injectivity, and surjectivity. Linearity means that the operator preserves addition and scalar multiplication. Injectivity means that different vectors in the input space are mapped to different equivalence classes in the quotient space. Surjectivity means that every equivalence class in the quotient space has at least one vector in the input space that maps to it.

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