General vector space question

In summary, The conversation discusses a problem involving real numbers and functions. Part 1 shows that composition by a mapping is a linear mapping from one set of functions to another. Part 2 explores the isomorphism of this mapping, specifically considering the cases of \phi being injective and surjective. To show that T is onto, you need to construct a function f in the set of functions from B to R such that T(f) equals a given function g from A to R. To show that T is one-to-one, you need to show that different inputs to T will always result in different outputs.
  • #1
fluxions
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Homework Statement


Sorry for the vague title!

Let R denote the set of real numbers, and F(S,R) denote the set of all functions from a set S to R.

Part 1: Let [tex]\phi[/tex] be any mapping from a set A to a set B. Show that composition by [tex]\phi[/tex] is a linear mapping from F(B,R) to F(A,R). That is, show that [tex] T : F(B,R) \rightarrow F(A,R) : f \mapsto f \circ \phi[/tex] is linear.

Part 2: In the situation given in part 1, show that T is an isomorphism if [tex]\phi[/tex] is bijective by show that:
(a) [tex]\phi[/tex] injective implies T surjective;
(b) [tex]\phi[/tex] surjective implies T injective.



The Attempt at a Solution


Well, I got part 1.
As for part 2... I have no clue. Any ideas?
 
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  • #2
OK, I will get you started on (a.) So we are supposing [itex]\phi[/itex] is 1-1. So for each a in A there is a unique b in B such that [itex]b = \phi(a)[/itex] and [itex]a =\phi^{-1}(b)[/itex]. Now suppose you are given g in F(A,R). To show T is onto, you need to build an f in F(B,R) such that T(f) = g, that is [itex]f\circ\phi = g[/itex].

For [itex]b \in \phi(A)[/itex], try defining [itex]f(b) = g(\phi^{-1}(b))[/itex]. Now see if you can show that T(f) = g, which is the same as showing [itex]f\circ\phi = g[/itex].

Also note, if b is exterior to [itex]\phi(A)[/itex], it doesn't matter what you define f(b).
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects, called vectors, and a set of operations, such as addition and scalar multiplication, that can be performed on those vectors. The vectors in a vector space can be real or complex numbers, but they can also be more abstract objects, such as functions or matrices.

2. What are the properties of a vector space?

There are several properties that a set of vectors must satisfy in order to be considered a vector space. These include closure under addition and scalar multiplication, associativity and commutativity of addition, distributivity of scalar multiplication over addition, and the existence of an additive identity element (the zero vector). Additionally, a vector space must have a set of axioms, such as the existence of inverses for addition and the existence of a multiplicative identity element for scalar multiplication.

3. How do you determine if a set of vectors form a vector space?

To determine if a set of vectors form a vector space, you must check if they satisfy all of the properties and axioms of a vector space. This includes checking if they are closed under addition and scalar multiplication, if they satisfy the associative, commutative, and distributive properties, and if they have an additive identity element and multiplicative identity element. If all of these conditions are met, then the set of vectors is a vector space.

4. What is the dimension of a vector space?

The dimension of a vector space is the number of vectors in a basis for that space. A basis is a set of linearly independent vectors that span the entire vector space. The dimension of a vector space can also be thought of as the minimum number of vectors needed to express any vector in that space. For example, the dimension of the vector space of 3-dimensional vectors is 3.

5. What are some real-world applications of vector spaces?

Vector spaces have many applications in various fields such as physics, engineering, computer science, and economics. They are used to represent physical quantities, such as velocity and force, in physics. In engineering, vector spaces are used to model and analyze systems. In computer science, vector spaces are used in machine learning and data analysis. In economics, vector spaces are used to model economic systems and analyze market trends.

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