Characterization of External Direct Sum - Cooperstein

In summary, Cooperstein explains that if a vector space V' has maps that satisfy properties (a) and (b), then it is isomorphic to the direct sum of the spaces V1 to Vn. He provides a formula for the isomorphism between V' and V, and its inverse, which can be shown to be linear and bijective.
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I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...

In Section 10.2 Cooperstein writes the following, essentially about external direct sums ... ...
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Cooperstein asserts that properties (a) and (b) above "characterize the space ##V## as the direct sum of the spaces ##V_1, \ ... \ ... \ , V_n##"

Can someone please explain how/why properties (a) and (b) above characterize the space ##V## as the direct sum of the spaces ##V_1, \ ... \ ... \ , V_n##?Help will be appreciated ...

Peter
 

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What he is saying is that, if we have a vector space ##V'## and for ##k=1,...,n## we have maps ##\epsilon_k':V_k\to V'## and ##\pi_k':V'\to V_k## that satisfy (a) and (b) then ##V'## is isomorphic to ##V## (which is the direct sum of ##V_1## to ##V_n##).

The isomorphism from ##V'## to ##V## is the map:

$$\vec v'\mapsto\sum_{k=1}^n\epsilon_k\pi_k'(\vec v')$$

and its inverse is the map

$$\vec v\mapsto\sum_{k=1}^n\epsilon_k'\pi_k(\vec v)$$

It is straightforward, if somewhat laborious, to show that this map is a bijection and that it is linear.
 
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HI Andrew ... reflecting on your post ...

Still trying to follow you ... but having some difficulty ...

Peter
 

1. What is External Direct Sum in the context of Cooperstein's characterization?

External Direct Sum, also known as Cartesian product, is a mathematical operation that combines two sets by pairing each element of one set with every element of the other set. In the context of Cooperstein's characterization, it refers to the combination of two mathematical objects, such as vector spaces, which results in a larger object with distinct properties.

2. How does External Direct Sum differ from Internal Direct Sum?

External Direct Sum differs from Internal Direct Sum in terms of the objects being combined. In External Direct Sum, two distinct objects are combined to form a new object, whereas in Internal Direct Sum, two subobjects of the same larger object are combined to form a new subobject.

3. What is the significance of Cooperstein's characterization of External Direct Sum?

Cooperstein's characterization provides a way to understand and analyze the properties of External Direct Sum. It helps to identify the conditions under which two objects can be combined to form an External Direct Sum, and also provides a way to prove the properties of the resulting object.

4. What are some applications of External Direct Sum in science and engineering?

External Direct Sum has many applications in science and engineering, particularly in fields such as linear algebra, abstract algebra, and group theory. It is used to study and solve problems related to vector spaces, matrices, and abstract algebraic structures. It also has applications in computer science, where it is used to represent complex data structures and perform operations on them efficiently.

5. Are there any limitations to Cooperstein's characterization of External Direct Sum?

While Cooperstein's characterization is a powerful tool for understanding and analyzing External Direct Sum, it does have some limitations. It may not apply to all mathematical objects, and some objects may have different properties that cannot be characterized using this approach. Additionally, the characterization may become more complex when dealing with more than two objects.

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