Characterization of External Direct Sum - Cooperstein

  • #1
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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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?temp_hash=392d5fdda952d6030121f7ddfed2887c.png




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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  • #2
andrewkirk
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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.
 
  • #3
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HI Andrew ... reflecting on your post ...

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

Peter
 

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