Question about isomorphic mapping on direct sums?

In summary, the identity map on the direct sum of V1 and V2 would be the composition of the inclusion map i1 and the projection map p1, plus the composition of the inclusion map i2 and the projection map p2. The same concept applies for an infinite direct sum and a direct product. However, in the case of an infinite index set, one must be careful about the definition of the sum. Ultimately, there is not much new content here, as this is already built into the definition of a direct sum.
  • #1
Ishida52134
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The identity map on the direct sum of V1 and V2 would be i1 composed with p1 + i2 composed with p2. Would such an identity map exist for an infinite direct sum? And an analogous mapping for a direct product?
 
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  • #2
You're given:
- [itex]V = \bigoplus_{a\in A} V_a[/itex], where [itex]A[/itex] is some possibly infinite index set.
- For each [itex]a\in A[/itex], the inclusion map [itex]\iota_a: V_a \to V[/itex] given by [itex]\iota_a(x_a) = (x_a, (0_b)_{b\in A\setminus\{a\}})[/itex].
- For each [itex]a\in A[/itex], the projection map [itex]\pi_a: V \to V_a[/itex] given by [itex]\pi_a(x) = (x_a)[/itex].

You've noticed that, if [itex]|A|<\infty[/itex], then [itex]\sum_{a\in A} \iota_a\circ\pi_a = id_V: V\to V[/itex].

So you're asking whether the condition that [itex]|A|<\infty[/itex] can be dropped to draw the same conclusion? The answer is essentially
1) Yes.
2) One has to be careful about the definition of [itex]\sum_{a\in A}[/itex] when [itex]A[/itex] is infinite.
3) There's very little content here, as point (2) is essentially built into the definition of a direct sum.
 

1. What is an isomorphic mapping?

An isomorphic mapping is a type of function that preserves the structure and properties of objects. In other words, it is a one-to-one and onto mapping that maintains the same relationships between elements in two different sets.

2. What is a direct sum?

A direct sum is a mathematical operation that combines two or more objects to create a new object. In the context of linear algebra, it is a way to combine vector spaces to create a larger vector space.

3. How do you perform an isomorphic mapping on direct sums?

To perform an isomorphic mapping on direct sums, you must first define the mapping between the two vector spaces. Then, you can apply the mapping to each element in the direct sum to create a new direct sum with the same structure and properties as the original.

4. What is the significance of isomorphic mapping on direct sums?

Isomorphic mapping on direct sums allows for the comparison and analysis of different vector spaces with similar structures. It also helps in solving problems related to linear transformations and finding equivalent representations of vector spaces.

5. Can an isomorphic mapping on direct sums preserve other properties besides structure?

Yes, an isomorphic mapping on direct sums can preserve other properties such as dimension, basis, and linear independence. This is because isomorphic mappings preserve the relationships between elements, not just the structure of the objects.

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