Word: Direct Sum
Part of Speech: Noun
Definition: In mathematics, a "direct sum" refers to a way of combining two groups (called sets) that do not share any elements (meaning they are disjoint). When you take the direct sum of these two sets, every element in the new set is made up of one element from each of the original sets added together.
Usage Instructions: You can use "direct sum" when discussing topics in mathematics, especially in areas like linear algebra, abstract algebra, or vector spaces. It is a more technical term and might not be commonly used outside of these subjects.
Imagine you have two sets: - Set A = {1, 2} - Set B = {3, 4}
In more advanced mathematics, "direct sum" can also refer to a method of combining vector spaces or modules. For instance, if you have two vector spaces, V and W, their direct sum is usually denoted as V ⊕ W. Each element in this direct sum can be represented as a pair (v, w) where v is from V and w is from W.
Outside of mathematics, "direct" can mean straightforward or clear without any confusion, while "sum" usually refers to the result of adding numbers. The combination "direct sum" in mathematics is specific to the idea of combining disjoint sets.
While "direct sum" is a specific term in math, similar concepts might include: - Sum: The result of adding numbers. - Combination: The act of joining together, though not necessarily disjoint.
There are no common idioms or phrasal verbs directly related to "direct sum" since it is a specialized term in mathematics.
Remember, "direct sum" is a mathematical concept for combining two different sets where every element is formed by adding an element from each set.