Problem 7. Let U1₁, U2,...,Um be subspaces of V, with m> 2. Prove that U₁+U₂+...+Um is a direct sum if and only if for each j = {1,2,...,m}, U₂n ([v₁) = {0}. i#j

Advanced Engineering Mathematics
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**Problem 7.** Let \( U_1, U_2, \ldots, U_m \) be subspaces of \( V \), with \( m \geq 2 \). Prove that \( U_1 + U_2 + \cdots + U_m \) is a direct sum if and only if for each \( j \in \{1, 2, \ldots, m\} \),

\[
U_j \cap \left( \sum_{i \neq j} U_i \right) = \{0\}.
\]
Transcribed Image Text:**Problem 7.** Let \( U_1, U_2, \ldots, U_m \) be subspaces of \( V \), with \( m \geq 2 \). Prove that \( U_1 + U_2 + \cdots + U_m \) is a direct sum if and only if for each \( j \in \{1, 2, \ldots, m\} \), \[ U_j \cap \left( \sum_{i \neq j} U_i \right) = \{0\}. \]
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