Sums of Subspaces: Is Addition Commutative & Associative?

  • Context: Graduate 
  • Thread starter Thread starter bjgawp
  • Start date Start date
  • Tags Tags
    Subspaces Sums
Click For Summary
SUMMARY

The addition of subspaces U_1, U_2, and U_3 of a vector space V over the fields R and C is both commutative and associative. This conclusion is derived from the properties of the elements within the subspaces, which maintain these operations. The summation of the subspaces results in the set of all possible sums of their elements, confirming that the addition of subspaces adheres to the same rules as the addition of their individual elements.

PREREQUISITES
  • Understanding of vector spaces and subspaces
  • Familiarity with the properties of commutativity and associativity
  • Knowledge of field theory, specifically over R and C
  • Basic concepts of linear combinations and sums of sets
NEXT STEPS
  • Study the properties of vector spaces in detail
  • Explore the implications of subspace addition in linear algebra
  • Learn about field theory and its applications in vector spaces
  • Investigate examples of commutative and associative operations in various mathematical structures
USEFUL FOR

Mathematicians, students of linear algebra, and anyone interested in the properties of vector spaces and subspaces.

bjgawp
Messages
84
Reaction score
0
If [tex]U_1, U_2, U_3,[/tex] are subspaces of V (over fields R and/or C), is the addition of the subspaces commutative and associative?

To me it seems rather trivial .. Since their summation is simply the set of all possible sums of the elements of [tex]U_1, U_2, U_3[/tex], and the elements themselves are associative and commutative, then so must be their subspaces and their sum.

Seems too easy to me ... I must be missing something ,,
 
Physics news on Phys.org
It is true. What you wrote is fine. But if you are doubtful, then you need to write it out more carefully.
 

Similar threads

Undergrad Understanding the concept of direct sum
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Confusion about the Direct Sum of Subspaces
  • · Replies 3 ·
Replies
3
Views
5K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
4K
MHB K^n as a K[T]-module - Exercise 1.2.8 (a) - Berrick and Keating (B&K)
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Near-Rings with Noncommutative Addition and Two-Sided Distributivity
  • · Replies 4 ·
Replies
4
Views
3K
MHB Vector Space - Proving Associativity
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Developing Intuition Behind Affine Subspaces: A Sample Problem
  • · Replies 6 ·
Replies
6
Views
4K
Graduate Question about the permutations of roots as polynomial coefficients
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad When is a subset a subspace in a vector space?
  • · Replies 7 ·
Replies
7
Views
3K
Dimension of orthogonal subspaces sum
  • · Replies 4 ·
Replies
4
Views
2K