Exercise: Given a collection of n vector spaces `V_1 , V_2 , ... , V_n` over a common field `bbb"F"` , we must:
1. Formally define the Direct Sum `V = V_1 o+ V_2 o+ ... o+ V_n` ("external" direct sum).
2. Define the operations of addition and scalar multiplication for this set.
3. Prove that this structure satisfies the vector space axioms , specifically that it inherits these properties from the individual spaces `V_i`.

Comment: This is a fundamental exercise. Understanding in general the Direct Sum , in this case the "external direct sum" , is crucial because , in Lie Algebra we often decompose complex algebras into a "Direct Sum" of simpler ones to make them easier to study.

General Strategy: The "Direct Sum" is essentiell the vector space version of a Cartesian product , also called the "External Direct Sum" in this case. Here we treat the elements of V as n-tuples , where each "slot" (e.g. x-slot, y-slot, z-slot , ...) in the tuple belongs to a different vector space. To prove it is a vector space , we will show that since each `V_i` individually obeys the rules of algebra , their "combined" form must also obey them component-wise.

Specific Theory Used:
- Definition of n-tuples: An element `v in V` is an ordered list `(v_1 \ , v_2 \ , ... \ , v_n)` where `v_i in V_i`.
- Vector Space Axioms: To be a vector space , the set must be closed under addition and scalar multiplication , and satisfy 8 axioms (associativity , commutativity , zero vector , additive inverse , and four distributive/identity laws for scalars).
- Component-wise Operations: Operations defined individually for each coordinate.

Solution:
Part A. The Definition: We define the Direct Sum `V=o+_(i = 1)^n V_i` as the set: `V = {(v_1 \ , v_2 \ , ... \ , v_n) : v_i in V_i \ "for all" \ i = 1 \ , ... \ , n}`.
We define the operations as follows:
- Addition: `(v_1 \ , ... \ , v_n) + (w_1 \ , ... \ , w_n) = (v_1 + w_1 \ , ... \ , v_n + w_n)`.
- Scalar Multiplication: `c(v_1 \ , ... \ , v_n) = (c v_1 \ , ... \ , c v_n)` for any `c in bbb"F"`.

Part B. Proving it is a Vector Space: To prove `V` is a vector space , we must verify the axioms. Let's check the most critical ones:
1. Closure: Since each `V_i` is a vector space , `v_i + w_i` is in `V_i` and `cv_i` is in `V_i`. Thus the resulting
n-tuple is still in `V`.
2. The zero vector: Each `V_i` has its own zero vector , `bb"0"_i`. We define the zero vector of `V` as `bb"0"_V = (bb"0"_1 \ , bb"0"_2 \ , ... \ , bb"0"_n)`. Adding this to any `v in V` yields `(v_1 + bb"0"_1 \ , ... \ , v_n + bb"0"_n) = (v_1 \ , ... \ , v_n) = v`.
3. Additive Inverse: For any `v = (v_1 \ , ... \ , v_n)` , the inverse is `- v = (-v_1 \ , ... \ , -v_n)`.
4. Distributivity: For `c in bbb"F" : c(v + w) = c(v_1 + w_1 \ , ... \ , v_n + w_n) = (c(v_1 + w_1) \ , ... \ , c(v_n + w_n))`
Using the the distributive law in each `V_i`:
`= (cv_1 + cw_1 \ , ... \ , cv_n + cw_n) = cv + cw`.
All other axioms (commutativity , associativity ) follow from the same component-wise logic.
QED