Characterization of External Direct Sum - Cooperstein

[Math Amateur]

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 ... ...

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

 

[andrewkirk]

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.

 

[Math Amateur]

HI Andrew ... reflecting on your post ...

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

Peter