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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:
In 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 ...
So ... 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:
In 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 ...
So ... 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
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