Given a set of [vector spaces](Vector%20spaces.md), $\{\mathcal{V}_1,... ,\mathcal{V_n}\},$ where $\mathcal{V}_1,..., \mathcal{V_n}$ are [subspaces](Vector%20spaces.md#Subspaces%20of%20vector%20spaces) of $\mathcal{W},$ the sum of these vector spaces $\mathcal{V}_1+...+ \mathcal{V_n}$ is defined as $\mathcal{V}_1+...+ \mathcal{V_n}=\{v_1+...+ v_n:v_1\in \mathcal{V}_1,..., v_n \in \mathcal{V}_n\}$ ^3c8b51
That is, the sum of vector spaces is the set of all possible sums of elements from $\{\mathcal{V}_1... \mathcal{V_n}\}.$
If $\mathcal{V}_1,..., \mathcal{V_n}$ is a subspace of $\mathcal{W},$ then the sum $\mathcal{V}_1+...+ \mathcal{V_n}$ is also a subspace of $\mathcal{W}$ due to closure under addition for elements of $\mathcal{V}_1+...+ \mathcal{V_n}$ ([proof](Sums%20of%20vector%20spaces.md#Proof%20that%20the%20sum%20of%20subspaces%20is%20also%20a%20subspace)).
If $\mathcal{V}_1,..., \mathcal{V_n}$ are subspaces of $\mathcal{V}=\mathcal{V}_1+...+ \mathcal{V_n},$ the elements in this vector space sum are each expressed as sums $v_1+...+ v_n$ where $v_i\in\mathcal{V_i}$ for each $i\in\{1,...,n\}.$ ^c00c25
%%Please do exercise 9 of chapter 1 of Linear Algebra done right!%%
# Direct sums of vector spaces
Consider a [sum of vector spaces](Sums%20of%20vector%20spaces.md), $\mathcal{V}=\mathcal{V}_1+...+ \mathcal{V_n},$
On way of showing that a [sum of vector spaces](Sums%20of%20vector%20spaces.md) is a [direct sum](Direct%20sums%20of%20vector%20spaces.md) is to show that
[(... see more)](Direct%20sums%20of%20vector%20spaces.md)
---
# Proofs and examples
## Proof that the sum of subspaces is also a subspace
---
# Recommended Reading
For an introduction to [sums of vector spaces](Sums%20of%20vector%20spaces.md) derived from the [Vector spaces](Vector%20spaces.md) axioms see:
* [Axler S., Gerhing F.W., Ribet K.A. _Linear Algebra Done Right_, Springer, 2nd edition, 1997](Axler%20S.,%20Gerhing%20F.W.,%20Ribet%20K.A.%20Linear%20Algebra%20Done%20Right,%20Springer,%202nd%20edition,%201997.md) pgs. 14 to 18.
For the relationship between direct sums of vector space and [Vector space dimension](Vector%20space%20dimension.md) see
* [Axler S., Gerhing F.W., Ribet K.A. _Linear Algebra Done Right_, Springer, 2nd edition, 1997](Axler%20S.,%20Gerhing%20F.W.,%20Ribet%20K.A.%20Linear%20Algebra%20Done%20Right,%20Springer,%202nd%20edition,%201997.md) pg 34. Here it is shown that if we find [property 3](Direct%20sums%20of%20vector%20spaces.md#^e70c9b) for a [sum of vector spaces](Sums%20of%20vector%20spaces.md) then we can deduce that this sum must also be a [direct sum.](Direct%20sums%20of%20vector%20spaces.md)
#MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces