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I am reading T. S. Blyth's book: Module Theory: An Approach to Linear Algebra ...
I am focused on Chapter 2: Submodules; intersections and sums ... and need help with the proof of Theorem 2.3 ...
Theorem 2.3 reads as follows:View attachment 8157In the above proof we read the following:
" ... ... A linear combination of elements of $$\bigcup_{ i \in I }$$ is precisely a sum of the form $$\sum_{ j \in J } m_j$$ for some $$J \in P(I).$$ ... ... "But ... Blyth defines a linear combination as in the text below ...https://www.physicsforums.com/attachments/8158So ... given the above definition wouldn't a linear combination of elements of $$\bigcup_{ i \in I } M_i$$ be a sum of the form $$\sum_{ j \in J } \lambda_j m_j$$ ... and not just $$\sum_{ j \in J } m_j$$ ... ... ?
Hope someone can help ...
Peter
I am focused on Chapter 2: Submodules; intersections and sums ... and need help with the proof of Theorem 2.3 ...
Theorem 2.3 reads as follows:View attachment 8157In the above proof we read the following:
" ... ... A linear combination of elements of $$\bigcup_{ i \in I }$$ is precisely a sum of the form $$\sum_{ j \in J } m_j$$ for some $$J \in P(I).$$ ... ... "But ... Blyth defines a linear combination as in the text below ...https://www.physicsforums.com/attachments/8158So ... given the above definition wouldn't a linear combination of elements of $$\bigcup_{ i \in I } M_i$$ be a sum of the form $$\sum_{ j \in J } \lambda_j m_j$$ ... and not just $$\sum_{ j \in J } m_j$$ ... ... ?
Hope someone can help ...
Peter