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Direct sum of two subspaces.

Let U1 and U2 be two subspaces of the same vector space V. The sum of these subspaces, denoted U1 + U2, is the set of all the sums $\overrightarrow{u_1} +\overrightarrow{u_2} $, where $\overrightarrow{u_1}\in U_1$ and $\overrightarrow{u_2}\in U_2$. If $U_1 \cap U_2 = \{ \overrightarrow{0}\}$, the sum is a direct sum and is denoted $U_1 \oplus U_2$. If $U_1 \oplus U_2=V$, then U1 and U2 are supplementary subspaces.

Example 5.5.1  
Take $U=\{ (x,0) \in \mathbb{R} ^2 \}$ and $V=\{ (0,y) \in \mathbb{R} ^2 \}$. We have: $U_1 \oplus U_2 = \mathbb{R} ^2$.
Let $\mathcal{F}(\mathbb{R} ,\mathbb{R} )$ be the vector space of all the functions defined on $\mathbb{R} $. Denote by $\mathcal{F}_{odd}$ (resp. $\mathcal{F}_{even}$) the subspace of all the odd (resp. even) functions from $\mathbb{R} $ to itself. Then $\mathcal{F}(\mathbb{R} ,\mathbb{R} ) = \mathcal{F}_{odd} \oplus \mathcal{F}_{even}$.

Figure 2: Direct sum of a plane and a line.
\mbox{\epsfig{file=directsum.eps,height=5cm} }

Proposition 5.5.2   Let U be a finite-dimensional space and let V1 and V2 be two subspaces such that $V_1 \oplus v_2=U$. Then

\begin{displaymath}\dim V_1 + \dim V_2 = \dim V.

This is linked to Thm  4.5.

Noah Dana-Picard