Sums of Subspaces: Is Addition Commutative & Associative?

In summary, a sum of subspaces is a mathematical operation where two or more subspaces are combined to create a new subspace. It is commutative, meaning the order of addition does not matter, and associative, meaning grouping does not affect the result. Subspaces with different dimensions cannot be added, and the sum of subspaces is related to linear combinations as it is the smallest subspace containing all possible combinations of the subspaces being added.
  • #1
bjgawp
84
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 ,,
 
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  • #2
It is true. What you wrote is fine. But if you are doubtful, then you need to write it out more carefully.
 

Related to Sums of Subspaces: Is Addition Commutative & Associative?

1. What is a sum of subspaces?

A sum of subspaces is a mathematical operation that combines two or more subspaces to create a new subspace. This operation is similar to addition in regular arithmetic, but instead of numbers, we are working with mathematical objects called subspaces which are subsets of a larger vector space.

2. Is addition of subspaces commutative?

Yes, addition of subspaces is commutative, which means that the order in which we add the subspaces does not matter. In other words, adding subspace A to subspace B will give the same result as adding subspace B to subspace A.

3. Is the sum of subspaces associative?

Yes, the sum of subspaces is associative, which means that when we are adding three or more subspaces, the grouping of the subspaces does not affect the final result. In other words, (A + B) + C = A + (B + C).

4. Can we add subspaces with different dimensions?

No, we cannot add subspaces with different dimensions. In order to add two subspaces, they must be of the same dimension. This is because subspaces are defined by a set of linearly independent vectors, and the dimension of a subspace is the number of vectors in this set.

5. How is the sum of subspaces related to linear combinations?

The sum of subspaces is closely related to linear combinations. This is because the sum of subspaces is the smallest subspace that contains all possible linear combinations of the subspaces being added. In other words, the sum of subspaces is the space spanned by the union of all the vectors in the subspaces being added.

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