affinely dependent set (english) - meaning, definition, translation, pronunciation

Part of Speech

• Noun Phrase

Phonetic Transcription

• /ˈæfɪnli dɪˈpɛndənt sɛt/

Meaning and Usage

An "affinely dependent set" refers to a collection of vectors in a vector space where at least one of the vectors can be expressed as an affine combination of the others. In simpler terms, this means that there is a linear relationship among the vectors in the set, which allows for one vector to be represented as a combination of the others with the constraint that the coefficients in the combination sum to one.

In the context of mathematics, particularly in linear algebra, affine dependency is a crucial concept. The frequency of use is more inclined towards written contexts, especially within academic texts, papers, and lectures discussing linear algebra and geometry.

Example Sentences

1. The mathematician proved that the given vectors formed an affinely dependent set, allowing for the simplification of calculations.

2. Математик доказал, что данные векторы образуют аффинно зависимый набор, что позволяет упростить расчёты.

3. In a three-dimensional space, if you have four points, they will always form an affinely dependent set.

4. В трёхмерном пространстве, если у вас есть четыре точки, они всегда будут образовывать аффинно зависимый набор.

5. To determine if the points form an affinely dependent set, one must check if they lie on the same plane.

6. Чтобы определить, образуют ли точки аффинно зависимый набор, необходимо проверить, лежат ли они в одной плоскости.

Idiomatic Expressions

Though "affinely dependent set” is not commonly found in idiomatic expressions, the concept of dependency can appear in mathematical phrases or in discussions about relationships and connections in various contexts.

1. “The arrangements were affinely dependent on previous performances.”

2. “Уст arrangements зависели от предыдущих выступлений.”

3. “The outcomes of the experiments were found to be affinely dependent, leading to a re-evaluation of the hypothesis.”

4. “Результаты экспериментов оказались аффинно зависимыми, что привело к пересмотру гипотезы.”

5. “A designer’s choices are often affinely dependent on the trends of the industry.”

6. “Выборы дизайнера часто аффинно зависимы от тенденций в индустрии.”

Etymology

The term "affinely" originates from "affine", which is derived from the Latin word "affinis", meaning "related by marriage" or "connected". In mathematical terms, it refers to a property of geometric relations, incorporating both linear and a zero vector combinations to describe dependencies.

Synonyms and Antonyms

• Synonyms: Dependent set, linearly dependent set (in a looser context, as affine dependence encompasses linear dependence).
• Antonyms: Affinely independent set, independent set.

This information encapsulates the essence of "affinely dependent set" and its applications in mathematical contexts.