Some thought about direct sum and

In summary, the condition that V is a direct sum of U and W means that V can be expressed as the sum of U and W, where there is no intersection between U and W. In some cases, this can also be stated as every element in V can be uniquely expressed as the sum of an element in U and an element in W. This is because U and W must be subspaces and therefore contain the zero vector, but there are no other shared elements between them.
  • #1
td21
Gold Member
177
8
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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  • #3
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.
 

1. What is a direct sum?

A direct sum is a mathematical operation that combines two or more vector spaces to form a new vector space. It is denoted by the symbol ⊕ and has properties similar to addition.

2. How is a direct sum different from a direct product?

In a direct sum, the combined vector space contains only elements that are composed of a single element from each of the original vector spaces. In a direct product, the combined vector space can contain elements that are composed of any combination of elements from the original vector spaces.

3. What is the significance of direct sums in linear algebra?

Direct sums are important in linear algebra because they allow us to break down a complex vector space into smaller, more manageable subspaces. This can simplify calculations and make it easier to understand the structure of a vector space.

4. Can a direct sum be infinite?

Yes, a direct sum can be infinite. In fact, direct sums are often used to combine an infinite number of vector spaces. For example, the space of all polynomials of degree at most n can be represented as a direct sum of n+1 copies of the space of all polynomials of degree n.

5. How is a direct sum related to direct summands?

A direct summand is a subspace of a vector space that, when combined with other direct summands, forms the original vector space. In other words, the vector space is the direct sum of its direct summands. This relationship is important in understanding the structure and decomposition of vector spaces.

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