Some thought about direct sum and

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SUMMARY

The discussion centers on the properties of direct sums in vector spaces, specifically addressing the conditions under which a vector space V can be expressed as a direct sum of subspaces U and W. It is established that for V to be a direct sum of U and W, it must hold that V = U + W and that the intersection of U and W is trivial, containing only the zero vector. Additionally, the unique representation of each vector v as a sum of vectors u from U and w from W is confirmed, emphasizing the importance of these conditions in linear algebra.

PREREQUISITES
  • Understanding of vector spaces and subspaces
  • Familiarity with the concept of direct sums in linear algebra
  • Knowledge of linear independence and span
  • Basic comprehension of unique representations in mathematical contexts
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  • Study the properties of vector space intersections and their implications
  • Explore the concept of linear independence in relation to direct sums
  • Learn about the implications of the zero vector in subspaces
  • Investigate the role of unique representations in other mathematical structures
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify the concept of direct sums and their properties.

td21
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I know that if V is a direct sum of U and W,
then
1. V=U+W
2 there is no intersection between U and W


However, in some books there is an equivalent condition:
3.Every v can be expressed uniquely as u+w


Why's that? Why can we be so sure about the word "unique"? Thanks.
 
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td21 said:
I know that if V is a direct sum of U and W,
then
1. V=U+W
2 there is no intersection between U and W

You aren't making a precise statement. Are U and W supspaces? What do you mean when you say "there is no intersection"? All subspaces contain the zero vector.
 

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