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In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g).
A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D.
Faithfulness means unfailingly remaining loyal to someone or something, and putting that loyalty into consistent practice regardless of extenuating circumstances. It may be exhibited, for example, by a husband or wife who does not engage in sexual relationships outside of the marriage . [ 1 ]
A typical sequence of secondary-school (grades 6 to 12) courses in mathematics reads: Pre-Algebra (7th or 8th grade), Algebra I, Geometry, Algebra II, Pre-calculus, and Calculus or Statistics. However, some students enroll in integrated programs [3] while many complete high school without passing Calculus or Statistics.
A spiritual test, according to several religious traditions, is a life situation, provided by God, to evaluate man's individual moral character and obedience to His laws. . Spiritual tests assess one's virtue in many aspects of everyday life, individual conduct, particularly when no witnesses exist, and reflect the maturation of the character with regards to such moral imperatives as honesty ...
The seven capital virtues or seven lively virtues (also known as the contrary or remedial virtues) [8] are those thought to stand in opposition to the seven capital vices (or deadly sins). Prudentius , writing in the 5th century, was the first author to allegorically represent Christian morality as a struggle between seven sins and seven virtues.
Fides (Latin: Fidēs) was the goddess of trust, faithfulness, and good faith in ancient Roman religion. [1] [2] Fides was one of the original virtues to be cultivated as a divinity with ceremonies and temples. [3] Fides Publica holding a cornucopia and extending a libation bowl on the reverse of a dupondius issued by Vespasian 77–78 AD
Mathematics textbooks are conventionally built up carefully, one chapter at a time, explaining what mathematicians would call the prerequisites before moving to a new topic. For example, you may think you can study Chapter 10 of a book before Chapter 9, but reading a few pages may then show you that you are wrong.