DIRECT SUM Meaning and
Definition
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A direct sum refers to a mathematical concept used in the field of linear algebra. Specifically, it denotes the sum of two or more vector spaces without any overlap, where the resulting vector space includes all possible combinations of vectors in the individual spaces.
In a direct sum, each vector from the combined space can be uniquely represented as a sum of vectors from the individual spaces. This decomposition property makes it distinct from an ordinary sum.
To define a direct sum, let V1, V2, …, Vn be vector spaces. Their direct sum, denoted as V1 ⊕ V2 ⊕ … ⊕ Vn, is constructed by taking the union of all vectors from these spaces, while ensuring that no vector is present in two or more spaces. Moreover, the resulting direct sum space retains the vector space structure.
Mathematically, if v1 ∈ V1, v2 ∈ V2, …, vn ∈ Vn, then any vector in the direct sum can be expressed as v = v1 + v2 + … + vn, where v ∈ V1 ⊕ V2 ⊕ … ⊕ Vn. Additionally, the direct sum requires that the sum be well-defined, meaning if v1 = v1' and v2 = v2', then v = v' as well.
The concept of a direct sum is essential in various branches of mathematics and physics, particularly in the study of linear transformations, subspaces, and decompositions of vector spaces. It provides a comprehensive and organized framework for analyzing and understanding vector spaces and their relationships.
Common Misspellings for DIRECT SUM
- sirect sum
- xirect sum
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- durect sum
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- ditect sum
- di5ect sum
- di4ect sum
- dirwct sum
Etymology of DIRECT SUM
The term "direct sum" comes from mathematics. In mathematics, the direct sum usually refers to the operation or concept of combining vectors or vector spaces.
The word "sum" is derived from Latin "summa", meaning "the highest, the top, the total". In mathematics, it generally signifies the result of adding two or more numbers or quantities.
The word "direct" is used to emphasize that the combining operation is done in a straightforward manner, without any interference or dependency between the components being combined. It signifies that the vectors or vector spaces are combined in a direct and non-overlapping way. This is in contrast to other types of sums or direct products where the components may interact or depend on each other in some way.
Collectively, "direct sum" in mathematics represents the straightforward, non-overlapping combination of vectors or vector spaces, without any interference or dependency.
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