Consider a [sum of vector spaces](Sums%20of%20vector%20spaces.md),   If the elements in the sum are uniquely represented by $v_1+...+ v_n$ for $v_i\in\mathcal{V}_i$ then the sum of vector spaces is referred to as a _direct sum of vector spaces._ The direct sum of vector spaces is then denoted by $\mathcal{V}=\mathcal{V}_1\oplus...\oplus \mathcal{V_n}$ ^6f7b6b # Properties of direct sums of vector spaces For a given [direct sum of vector spaces,](Direct%20sums%20of%20vector%20spaces.md) $\mathcal{V}=\mathcal{V}_1\oplus...\oplus \mathcal{V_n}$ where $v_1+...+ v_n \in\mathcal{V}$ 1. The only sum, $v_1+...+ v_n,$ that represents the $0$ element is one where all the vectors in the sum are equal to $0$^9c444d 2. Consider the case where $\mathcal{U}$ and $\mathcal{W}$ are [subspaces](Vector%20spaces.md#Subspaces%20of%20vector%20spaces) of $\mathcal{V}$ and $\mathcal{V}=\mathcal{U}\oplus\mathcal{W},$ then $\mathcal{U}\cap\mathcal{W}=$[$\{0\}$](Vector%20spaces.md#The%20zero%20vector%20space). 3. [$\mathrm{dim}$](Vector%20space%20dimension.md)$(\mathcal{V})=\mathrm{dim}(\mathcal{V}_1)+...+\mathrm{dim}(\mathcal{V}_n)$ ^e70c9b [property 1](Direct%20sums%20of%20vector%20spaces.md#^9c444d) follows from [uniqueness](Direct%20sums%20of%20vector%20spaces.md#^6f7b6b) in the definition of [direct sums of vector spaces.](Direct%20sums%20of%20vector%20spaces.md) %%Here we need to prove these properties the proofs of these properties should all be in Axler's textbook%% %%Given [[Real vector spaces]]s $\mathcal{U}$ and $\mathcal{V}$ their[](Direct%20sum.md)]]_ is defined by the following [[Sets]] of ordered pairs that itself also a real vector space $\mathcal{V}\oplus\mathcal{U}=\{(u,v)|u\in\mathcal{U}, v\in\mathcal{V}\}$%% # Orthogonal Complement of a vector space [(... see more)](Orthogonal%20complement%20of%20a%20subspace) %%See page 111 of Linear Algebra done right%% # Operators on direct sums of vector spaces %%See page 76 of Linear Algebra done right in the chapter on eigenvalues and eigenvectors%% --- # Proofs and examples %%Prove property 2. The proof also requires the completion of exercise 1.8%% --- # Recommended Reading For an introduction to the notion of a [direct sum of vector spaces](Direct%20sums%20of%20vector%20spaces.md) as it is derived from the notion of [sums of vector spaces](Sums%20of%20vector%20spaces.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) pgs. 14 to 18. What this section conveys as well is the notion that sums of vector spaces are analogous to unions of sets while direct sums are analogous to disjoint unions as we convert from the logic of sets to the logic of vector spaces. The important idea that drives the structure of this conversion is the vector space axioms. This text is an introduction to linear algebra aimed at undergraduates studying mathematics. 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