-
Given two vector subspaces \((\mathcal{U},
\mathcal{S}, +)\), \((\mathcal{V},
\mathcal{S}, +)\) of the space \((\mathcal{W},
\mathcal{S}, +)\), the sum is the set
\(\mathcal{U}+\mathcal{V}= \{
\boldsymbol{u}+\boldsymbol{v} | \boldsymbol{u} \in
\mathcal{U},
\boldsymbol{v} \in \mathcal{V} \} .\)
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Given two vector subspaces \((\mathcal{U},
\mathcal{S}, +)\), \((\mathcal{V},
\mathcal{S}, +)\) of the space \((\mathcal{W},
\mathcal{S}, +)\), the direct sum is
the set \(\mathcal{U} \oplus \mathcal{V}= \{
\boldsymbol{u}+\boldsymbol{v} | \exists
!\boldsymbol{u} \in
\mathcal{U}, \exists !\boldsymbol{v} \in \mathcal{V} \} .\)
(unique decomposition)
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Given two vector subspaces \((\mathcal{U},
\mathcal{S}, +)\), \((\mathcal{V},
\mathcal{S}, +)\) of the space \((\mathcal{W},
\mathcal{S}, +)\), the intersection is
the set
\(\displaystyle \mathcal{U} \cap \mathcal{V}= \{ \boldsymbol{x} |
\boldsymbol{x} \in
\mathcal{U}, \boldsymbol{x} \in \mathcal{V} \}
.\)
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Two vector subspaces \((\mathcal{U}, \mathcal{S},
+)\), \((\mathcal{V}, \mathcal{S}, +)\)
of the space \((\mathcal{W}, \mathcal{S}, +)\)
are orthogonal subspaces, denoted \(\mathcal{U}
\bot \mathcal{V}\) if \(\boldsymbol{u}^T
\boldsymbol{v}= 0\) for any \(\boldsymbol{u}
\in \mathcal{U}, \boldsymbol{v} \in \mathcal{V}\).
-
Two vector subspaces \((\mathcal{U}, \mathcal{S},
+)\), \((\mathcal{V}, \mathcal{S}, +)\)
of the space \((\mathcal{W}, \mathcal{S}, +)\)
are orthogonal complements, denoted \(\mathcal{U}=\mathcal{V}^{\bot}\), \(\mathcal{V}=\mathcal{U}^{\bot}\)
if \(\mathcal{U} \bot \mathcal{V}\) and \(\mathcal{U}+\mathcal{V}=\mathcal{W}\).
-
Orthogonal complement subspaces form a direct sum \(\mathcal{U}=\mathcal{V}^{\bot}\),
\(\mathcal{V}=\mathcal{U}^{\bot} \Rightarrow\)
\(\displaystyle \mathcal{U}+\mathcal{V}=\mathcal{U} \oplus
\mathcal{V}\)
