Linear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
Linear formIn mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered.
CarbonCarbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent—its atom making four electrons available to form covalent chemical bonds. It belongs to group 14 of the periodic table. Carbon makes up about 0.025 percent of Earth's crust. Three isotopes occur naturally, ^12C and ^13C being stable, while ^14C is a radionuclide, decaying with a half-life of about 5,730 years. Carbon is one of the few elements known since antiquity.
Chemical formulaIn chemistry, a chemical formula is a way of presenting information about the chemical proportions of atoms that constitute a particular chemical compound or molecule, using chemical element symbols, numbers, and sometimes also other symbols, such as parentheses, dashes, brackets, commas and plus (+) and minus (−) signs. These are limited to a single typographic line of symbols, which may include subscripts and superscripts. A chemical formula is not a chemical name since it does not contain any words.
Molecular massThe molecular mass (m) is the mass of a given molecule, for which the unit dalton (Da) is used. Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The related quantity relative molecular mass, as defined by IUPAC, is the ratio of the mass of a molecule to the atomic mass constant (which is equal to one dalton) and is unitless. The molecular mass and relative molecular mass are distinct from but related to the molar mass.
Quaternary ammonium cationIn chemistry, quaternary ammonium cations, also known as quats, are positively-charged polyatomic ions of the structure , where R is an alkyl group, an aryl group or organyl group. Unlike the ammonium ion () and the primary, secondary, or tertiary ammonium cations, the quaternary ammonium cations are permanently charged, independent of the pH of their solution. Quaternary ammonium salts or quaternary ammonium compounds (called quaternary amines in oilfield parlance) are salts of quaternary ammonium cations.
Linear algebraLinear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Trivial groupIn mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: or depending on the context. If the group operation is denoted then it is defined by The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group.
PolysaccharidePolysaccharides (ˌpɒliˈsækəraɪd), or polycarbohydrates, are the most abundant carbohydrates found in food. They are long-chain polymeric carbohydrates composed of monosaccharide units bound together by glycosidic linkages. This carbohydrate can react with water (hydrolysis) using amylase enzymes as catalyst, which produces constituent sugars (monosaccharides, or oligosaccharides). They range in structure from linear to highly branched. Examples include storage polysaccharides such as starch, glycogen and galactogen and structural polysaccharides such as cellulose and chitin.
AlkaneIn organic chemistry, an alkane, or paraffin (a historical trivial name that also has other meanings), is an acyclic saturated hydrocarbon. In other words, an alkane consists of hydrogen and carbon atoms arranged in a tree structure in which all the carbon–carbon bonds are single. Alkanes have the general chemical formula . The alkanes range in complexity from the simplest case of methane (), where n = 1 (sometimes called the parent molecule), to arbitrarily large and complex molecules, like pentacontane () or 6-ethyl-2-methyl-5-(1-methylethyl) octane, an isomer of tetradecane ().
