> [!NOTE] Definition (Direct sum of vector spaces) > Let $V$ and $W$ be [[Vector spaces|vector spaces]] over the same field. The **direct sum** of $V$ and $W$ is the [[Vector spaces|vector space]] $V\oplus W = \{ (v,w) \mid v\in V, w\in W \}$(the [[Cartesian Product|cartesian product]] $V\times W$) where the vector space operations are $\begin{align} (v_{1},w_{1}) + (v_{2},w_{2}) & = (v_{1}+v_{2},w_{1}+w_{2}) \\ \lambda(v,w) &= (\lambda v, \lambda w) \\ \end{align}$where the left-hand side of each line is the operation in $V \oplus W$ , and it is defined by the right-hand side, which only involves operations in $V$ or $W$. > >The additive identity is $0_{V \oplus W}=(0_{V},0_{W})$, and the additive inverse is $-(v,w)=(-v,-w).$ # Properties > [!NOTE] Proposition (Basis of direct sum) > If $V,W$ are [[Vector spaces|finite dimensional vector spaces]] over $\mathbb{R}$ and $B_{V} = \{ v_{1},\dots v_{s} \}$ is a [[Basis of Vector Space|basis]] of $V$, $B_{W}= \{ w_{1},\dots,w_{t} \}$ is a *basis* of $W$. Then $B = (B_{V}\times \{ 0_{W} \} ) \cup (\{ 0_{V} \} \times B_{W})$is a basis of $V\oplus W$. >*Proof.* If $(v,w)\in V \oplus W$ then by finite dimensionality of $V$ and $W$, there are scalars $\lambda_{i},\mu_{j}\in \mathbb{R}$ for which $v=\lambda_{1}v_{1}+\dots\lambda_{s} v_{s} \quad \text{ and} \quad w = \mu_{1}w_{1}+\dots \mu_{t}w_{t}$and therefore $B$ spans since $(v,w) = \lambda_{1}(v_{1},0_{W})+\dots\lambda_{s} (v_{s},0_{W}) +\dots +\mu_{1} (0_{V}, w_{1})+\dots \mu_{t} (0_{V}, w_{t}).$if $\lambda_{i},\mu_{j}\in\mathbb{R}$ $\lambda_{1}(v_{1},0_{W})+\dots +\lambda_{s}(v_{s},0_{W})+ \mu_{1}(0_{V},w_{1})+\dots+ \mu_{t}(0_{V}, w_{t}) = 0_{V\oplus W}$then $\lambda_{1}v_{1}+\dots\lambda_{s} v_{s}=0_{V}$ and $\mu_{1}w_{1}+\dots \mu_{t}w_{t}=0_{W}$ and so all $\lambda_{i}=\mu_{j}=0$ since $B_{V}, B_{W}$ are linearly independent. $\square$ > [!NOTE] Corollary (Dimension of direct sum) > If $V,W$ are [[Vector spaces|finite dimensional]] then $\dim V \oplus W = \dim V + \dim W$ > [!Example] > Let $V=\mathbb{R}^{2}$ and $W=\mathbb{R}[x]_{\leq 2}$. Then $V\oplus W=\{ ( (\lambda_{1}, \lambda_{2})^{T}, \lambda_{3} +\lambda_{4} x + \lambda_{5} x^{2}) \mid \lambda_{1}, \dots \lambda_{5} \in \mathbb{R} \}.$Choose [[Basis of Vector Space|basis]] $(1,0)^{T}, (0,1)$ of $V$, $1,x^{2},x^{3}$ of $W$. Then $((1,0)^{T}, 0), \; ((0,1)^{T},0), \; (0,1), \; (0,x), \; (0,x^{2})$is a basis of $V \oplus W$ which we will call $v_{1},v_{2},v_{3},v_{4},v_{5}$ respectively so that $V \oplus W = \{ \lambda_{1}v_{1} +\dots + \lambda_{5} v_{5} \mid \lambda_{1},\dots,\lambda_{5} \in \mathbb{R} \}$