Linear Algebra: Operations with Vector Spaces

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SUMMARY

The discussion focuses on operations within vector spaces, specifically subspace addition and intersection in the context of a vector space V over a field k. It establishes that there exists a zero element in the set of subspaces S for subspace addition, confirming its associative property. Additionally, the intersection operation is shown to be associative, although the existence of an identity element for intersection remains uncertain. The conversation highlights confusion regarding the definitions and operations related to subspaces.

PREREQUISITES
  • Understanding of vector spaces and subspaces
  • Familiarity with operations on vector spaces, specifically addition and intersection
  • Knowledge of associative properties in algebraic structures
  • Basic concepts of linear algebra, including zero elements in vector spaces
NEXT STEPS
  • Study the properties of vector space operations in detail
  • Learn about the direct sum of vector spaces and its implications
  • Research the concept of identity elements in algebraic structures
  • Examine examples of subspace addition and intersection in various vector spaces
USEFUL FOR

Students of linear algebra, educators teaching vector space concepts, and anyone seeking to deepen their understanding of operations within vector spaces.

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Homework Statement



Let V be a vector space over k and S the set of all subspaces of V. Consider the operation of subspace addition in S. Show that there is a zero in S for this operation and that the operation is associative. Consider the operation of intersection in S. Show that this operation is associative. Is there an identity for this operation (i.e., there is an E existing in S such that A intersect E = A for all E in S)?

Homework Equations





The Attempt at a Solution



Let U and W be in S. Then U+W=W+U=0+0. U intersect W = W intersect U...I'm not really sure where else to go with this...
 
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What do you mean by "subspace addition"? The direct product? You say "U+ W= W+ U= 0+ 0". What is "0"?
 
HallsofIvy said:
What do you mean by "subspace addition"? The direct product? You say "U+ W= W+ U= 0+ 0". What is "0"?


I guess the 0 that E U and the 0 that E W? You know, I'm not really sure. We're not using a published book for my Linear Algebra class. We have pdf pages of the professor's lecture notes but they are really short (very minimal explanations), full of typos, and examples without solutions. To be honest, I'm not really sure even what the question is asking. =/
 

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