Vector (mathematics and physics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)

In mathematics and physics, vector is a term that refers informally to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.

Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.

The term vector is also used, in some contexts, for tuples, which are finite sequences (of numbers or other objects) of a fixed length.

Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called a coordinate vector space.

Many vector spaces are considered in mathematics, such as extension fields, polynomial rings, algebras and function spaces. The term vector is generally not used for elements of these vector spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces (for example, when discussing general properties of vector spaces).

Vectors in Euclidean geometry

Main article: Euclidean vector

A vector pointing from A to B

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by $\stackrel{⟶}{AB}.$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.

Vector spaces

Main article: Vector space

Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars: real coordinate space or complex coordinate space.

Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations.

Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension.

Many vector spaces that are considered in mathematics are also endowed with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras and Lie algebras. This is also the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces.

Vectors in algebra

Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly due to historical reasons.

• Vector quaternion, a quaternion with a zero real part
• Multivector or p-vector, an element of the exterior algebra of a vector space.
• Spinors, also called spin vectors, have been introduced for extending the notion of rotation vector. In fact, rotation vectors represent well rotations locally, but not globally, because a closed loop in the space of rotation vectors may induce a curve in the space of rotations that is not a loop. Also, the manifold of rotation vectors is orientable, while the manifold of rotations is not. Spinors are elements of a vector subspace of some Clifford algebra.
• Witt vector, an infinite sequence of elements of a commutative ring, which belongs to an algebra over this ring, and has been introduced for handling carry propagation in the operations on p-adic numbers.

Data represented by vectors

See also: Vector (data type)

The set ${\displaystyle {\mathbb{R}}^n}$ of tuples of n real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication. It is common to call these tuples vectors, even in contexts where vector-space operations do not apply. More generally, when some data can be represented naturally by vectors, they are often called vectors even when addition and scalar multiplication of vectors are not valid operations on these data. Here are some examples.

• Rotation vector, a Euclidean vector whose direction is that of the axis of a rotation and magnitude is the angle of the rotation.
• Burgers vector, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
• Interval vector, in musical set theory, an array that expresses the intervallic content of a pitch-class set
• Probability vector, in statistics, a vector with non-negative entries that sum to one.
• Random vector or multivariate random variable, in statistics, a set of real-valued random variables that may be correlated. However, a random vector may also refer to a random variable that takes its values in a vector space.
• Logical vector, a vector of 0s and 1s (Booleans).

Vectors in calculus

Calculus serves as a foundational mathematical tool in the realm of vectors, offering a framework for the analysis and manipulation of vector quantities in diverse scientific disciplines, notably physics and engineering. Vector-valued functions, where the output is a vector, are scrutinized using calculus to derive essential insights into motion within three-dimensional space. Vector calculus extends traditional calculus principles to vector fields, introducing operations like gradient, divergence, and curl, which find applications in physics and engineering contexts. Line integrals, crucial for calculating work along a path within force fields, and surface integrals, employed to determine quantities like flux, illustrate the practical utility of calculus in vector analysis. Volume integrals, essential for computations involving scalar or vector fields over three-dimensional regions, contribute to understanding mass distribution, charge density, and fluid flow rates.

Scalar (mathematics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Scalar_(mathematics)

A scalar is an element of a field which is used to define a vector space. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers).

A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space.

A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector. The term scalar is also sometimes used informally to mean a vector, matrix, tensor, or other, usually, "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1 × n matrix and an n × 1 matrix, which is formally a 1 × 1 matrix, is often said to be a scalar. The real component of a quaternion is also called its scalar part.

The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.

Etymology

The word scalar derives from the Latin word scalaris, an adjectival form of scala (Latin for "ladder"), from which the English word scale also comes. The first recorded usage of the word "scalar" in mathematics occurs in François Viète's Analytic Art (In artem analyticem isagoge) (1591):

Magnitudes that ascend or descend proportionally in keeping with their nature from one kind to another may be called scalar terms.

(Latin: Magnitudines quae ex genere ad genus sua vi proportionaliter adscendunt vel descendunt, vocentur Scalares.)

According to a citation in the Oxford English Dictionary the first recorded usage of the term "scalar" in English came with W. R. Hamilton in 1846, referring to the real part of a quaternion:

The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity; we shall call it therefore the scalar part.

Definitions and properties

Scalars are real numbers used in linear algebra, as opposed to vectors. This image shows a Euclidean vector. Its coordinates x and y are scalars, as is its length, but v is not a scalar.

Scalars of vector spaces

A vector space is defined as a set of vectors (additive abelian group), a set of scalars (field), and a scalar multiplication operation that takes a scalar k and a vector v to form another vector kv. For example, in a coordinate space, the scalar multiplication ${\displaystyle k(v_1,v_2,\dots ,v_n)}$ yields ${\displaystyle (k{v}_1,k{v}_2,\dots ,k{v}_n)}$. In a (linear) function space, kf is the function x ↦ k(f(x)).

The scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields.

Scalars as vector components

According to a fundamental theorem of linear algebra, every vector space has a basis. It follows that every vector space over a field K is isomorphic to the corresponding coordinate vector space where each coordinate consists of elements of K (E.g., coordinates (a₁, a₂, ..., a_n) where a_i ∈ K and n is the dimension of the vector space in consideration.). For example, every real vector space of dimension n is isomorphic to the n-dimensional real space Rⁿ.

Scalars in normed vector spaces

Alternatively, a vector space V can be equipped with a norm function that assigns to every vector v in V a scalar ||v||. By definition, multiplying v by a scalar k also multiplies its norm by |k|. If ||v|| is interpreted as the length of v, this operation can be described as scaling the length of v by k. A vector space equipped with a norm is called a normed vector space (or normed linear space).

The norm is usually defined to be an element of V's scalar field K, which restricts the latter to fields that support the notion of sign. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as the four arithmetic operations; thus the rational numbers Q are excluded, but the surd field is acceptable. For this reason, not every scalar product space is a normed vector space.

Scalars in modules

When the requirement that the set of scalars form a field is relaxed so that it need only form a ring (so that, for example, the division of scalars need not be defined, or the scalars need not be commutative), the resulting more general algebraic structure is called a module.

In this case the "scalars" may be complicated objects. For instance, if R is a ring, the vectors of the product space Rⁿ can be made into a module with the n×n matrices with entries from R as the scalars. Another example comes from manifold theory, where the space of sections of the tangent bundle forms a module over the algebra of real functions on the manifold.

Scaling transformation

The scalar multiplication of vector spaces and modules is a special case of scaling, a kind of linear transformation.

Chemical equation

From Wikipedia, the free encyclopedia

A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas. The reactant entities are given on the left-hand side and the product entities are on the right-hand side with a plus sign between the entities in both the reactants and the products, and an arrow that points towards the products to show the direction of the reaction. The chemical formulas may be symbolic, structural (pictorial diagrams), or intermixed. The coefficients next to the symbols and formulas of entities are the absolute values of the stoichiometric numbers. The first chemical equation was diagrammed by Jean Beguin in 1615.

Structure

A chemical equation (see an example below) consists of a list of reactants (the starting substances) on the left-hand side, an arrow symbol, and a list of products (substances formed in the chemical reaction) on the right-hand side. Each substance is specified by its chemical formula, optionally preceded by a number called stoichiometric coefficient. The coefficient specifies how many entities (e.g. molecules) of that substance are involved in the reaction on a molecular basis. If not written explicitly, the coefficient is equal to 1. Multiple substances on any side of the equation are separated from each other by a plus sign.

As an example, the equation for the reaction of hydrochloric acid with sodium can be denoted:

2HCl + 2Na → 2NaCl + H₂

Given the formulas are fairly simple, this equation could be read as "two H-C-L plus two N-A yields^[b] two N-A-C-L and H two." Alternately, and in general for equations involving complex chemicals, the chemical formulas are read using IUPAC nomenclature, which could verbalise this equation as "two hydrochloric acid molecules and two sodium atoms react to form two formula units of sodium chloride and a hydrogen gas molecule."

Reaction types

Different variants of the arrow symbol are used to denote the type of a reaction:

${\displaystyle \to }$	net forward reaction
${\displaystyle \rightleftarrows }$	reaction in both directions[c]
${\displaystyle \rightleftharpoons }$	equilibrium[d]
${\displaystyle =}$	stoichiometric relation
${\displaystyle \leftrightarrow }$	resonance (not a reaction)

State of matter

To indicate physical state of a chemical, a symbol in parentheses may be appended to its formula: (s) for a solid, (l) for a liquid, (g) for a gas, and (aq) for an aqueous solution. This is especially done when one wishes to emphasize the states or changes thereof. For example, the reaction of aqueous hydrochloric acid with solid (metallic) sodium to form aqueous sodium chloride and hydrogen gas would be written like this:

2HCl(aq) + 2Na(s) → 2NaCl(aq) + H₂(g)

That reaction would have different thermodynamic and kinetic properties if gaseous hydrogen chloride were to replace the hydrochloric acid as a reactant:

2HCl(g) + 2Na(s) → 2NaCl(s) + H₂(g)

Alternately, an arrow without parentheses is used in some cases to indicate formation of a gas ↑ or precipitate ↓. This is especially useful if only one such species is formed. Here is an example indicating that hydrogen gas is formed:

2HCl + 2Na → 2 NaCl + H₂ ↑

Catalysis and other conditions

The Baker–Venkataraman rearrangement needs a base as a catalyst.

If the reaction requires energy, it is indicated above the arrow. A capital Greek letter delta (Δ) or a triangle (△)^[e] is put on the reaction arrow to show that energy in the form of heat is added to the reaction. The expression hν^[f] is used as a symbol for the addition of energy in the form of light. Other symbols are used for other specific types of energy or radiation.

Similarly, if a reaction requires a certain medium with certain specific characteristics, then the name of the acid or base that is used as a medium may be placed on top of the arrow. If no specific acid or base is required, another way of denoting the use of an acidic or basic medium is to write H⁺ or OH⁻ (or even "acid" or "base") on top of the arrow. Specific conditions of the temperature and pressure, as well as the presence of catalysts, may be indicated in the same way.

Notation variants

This illustration of a mechanism for acid-catalyzed hydrolysis of an amide puts some reactants and products above the arrows and to their branches to allow chaining of the chemical "equations".

The standard notation for chemical equations only permits all reactants on one side, all products on the other, and all stoichiometric coefficients positive. For example, the usual form of the equation for dehydration of methanol to dimethylether is:

2 CH₃OH → CH₃OCH₃ + H₂O

Sometimes an extension is used, where some substances with their stoichiometric coefficients are moved above or below the arrow, preceded by a plus sign or nothing for a reactant, and by a minus sign for a product. Then the same equation can look like this:

${\displaystyle 2{\text{CH}}_3^{}\text{OH}\stackrel{\stackrel{}{-{\text{H}}_2^{}\text{O}}}{\to }{\text{CH}}_3^{}{\text{OCH}}_3^{}}$

Such notation serves to hide less important substances from the sides of the equation, to make the type of reaction at hand more obvious, and to facilitate chaining of chemical equations. This is very useful in illustrating multi-step reaction mechanisms. Note that the substances above or below the arrows are not catalysts in this case, because they are consumed or produced in the reaction like ordinary reactants or products.

Another extension used in reaction mechanisms moves some substances to branches of the arrow. Both extensions are used in the example illustration of a mechanism.

Use of negative stoichiometric coefficients at either side of the equation (like in the example below) is not widely adopted and is often discouraged.

${\displaystyle 2{\text{CH}}_3^{}\text{OH}-{\text{H}}_2^{}\text{O}⟶{\text{CH}}_3^{}{\text{OCH}}_3^{}}$

Balancing chemical equations

Main article: Stoichiometry

Because no nuclear reactions take place in a chemical reaction, the chemical elements pass through the reaction unchanged. Thus, each side of the chemical equation must represent the same number of atoms of any particular element (or nuclide, if different isotopes are taken into account). The same holds for the total electric charge, as stated by the charge conservation law. An equation adhering to these requirements is said to be balanced.

A chemical equation is balanced by assigning suitable values to the stoichiometric coefficients. Simple equations can be balanced by inspection, that is, by trial and error. Another technique involves solving a system of linear equations.

Balanced equations are usually written with smallest natural-number coefficients. Yet sometimes it may be advantageous to accept a fractional coefficient, if it simplifies the other coefficients. The introductory example can thus be rewritten as

${\displaystyle \text{HCl}+\text{Na}⟶\text{NaCl}+\frac{1}{2}{\text{H}}_2^{}}$

In some circumstances the fractional coefficients are even inevitable. For example, the reaction corresponding to the standard enthalpy of formation must be written such that one molecule of a single product is formed. This will often require that some reactant coefficients be fractional, as is the case with the formation of lithium fluoride:

${\displaystyle \text{Li}(\text{s})+\frac{1}{2}{\text{F}}_2^{}(\text{g})⟶\text{LiF}(\text{s})}$

Inspection method

As seen from the equation CH
₄ + 2 O
₂ → CO
₂ + 2 H
₂O, a coefficient of 2 must be placed before the oxygen gas on the reactants side and before the water on the products side in order for, as per the law of conservation of mass, the quantity of each element does not change during the reaction

P₄O₁₀ + 6 H₂O → 4 H₃PO₄
This chemical equation is being balanced by first multiplying H₃PO₄ by four to match the number of P atoms, and then multiplying H₂O by six to match the numbers of H and O atoms.

The method of inspection can be outlined as setting the most complex substance's stoichiometric coefficient to 1 and assigning values to other coefficients step by step such that both sides of the equation end up with the same number of atoms for each element. If any fractional coefficients arise during this process, the presence of fractions may be eliminated (at any time) by multiplying all coefficients by their lowest common denominator.

Example

Balancing of the chemical equation for the complete combustion of methane

${\displaystyle ?{\text{CH}}_4^{}+?{\text{O}}_2^{}⟶?{\text{CO}}_2^{}+?{\text{H}}_2^{}\text{O}}$

is achieved as follows:

1. A coefficient of 1 is placed in front of the most complex formula (CH₄):

   ${\displaystyle 1{\text{CH}}_4^{}+?{\text{O}}_2^{}⟶?{\text{CO}}_2^{}+?{\text{H}}_2^{}\text{O}}$

2. The left-hand side has 1 carbon atom, so 1 molecule of CO₂ will balance it. The left-hand side also has 4 hydrogen atoms, which will be balanced by 2 molecules of H₂O:

   ${\displaystyle 1{\text{CH}}_4^{}+?{\text{O}}_2^{}⟶1{\text{CO}}_2^{}+2{\text{H}}_2^{}\text{O}}$

3. Balancing the 4 oxygen atoms of the right-hand side by 2 molecules of O₂ yields the equation

   ${\displaystyle 1{\text{CH}}_4^{}+2{\text{O}}_2^{}⟶1{\text{CO}}_2^{}+2{\text{H}}_2^{}\text{O}}$

4. The coefficients equal to 1 are omitted, as they do not need to be specified explicitly:

   ${\displaystyle {\text{CH}}_4^{}+2{\text{O}}_2^{}⟶{\text{CO}}_2^{}+2{\text{H}}_2^{}\text{O}}$

5. It is wise to check that the final equation is balanced, i.e. that for each element there is the same number of atoms on the left- and right-hand side: 1 carbon, 4 hydrogen, and 4 oxygen.

System of linear equations

For each chemical element (or nuclide or unchanged moiety or charge) i, its conservation requirement can be expressed by the mathematical equation

${\displaystyle \sum _{j\in \text{reactants}}{a}_{ij}{s}_j=\sum _{j\in \text{products}}{a}_{ij}{s}_j}$

where

a_ij is the number of atoms of element i in a molecule of substance j (per formula in the chemical equation), and

s_j is the stoichiometric coefficient for the substance j.

This results in a homogeneous system of linear equations, which are readily solved using mathematical methods. Such system always has the all-zeros trivial solution, which we are not interested in, but if there are any additional solutions, there will be infinite number of them. Any non-trivial solution will balance the chemical equation. A "preferred" solution is one with whole-number, mostly positive stoichiometric coefficients s_j with greatest common divisor equal to one.

Example

Let us assign variables to stoichiometric coefficients of the chemical equation from the previous section and write the corresponding linear equations:

${\displaystyle {\mathit{\text{s}}}_1^{}{\text{CH}}_4^{}+{\mathit{\text{s}}}_2^{}{\text{O}}_2^{}⟶{\mathit{\text{s}}}_3^{}{\text{CO}}_2^{}+{\mathit{\text{s}}}_4^{}{\text{H}}_2^{}\text{O}}$

${\displaystyle \begin{array}{rlrl}\text{C:}& & s_1& =s_3\\ \text{H:}& & 4s_1& =2s_4\\ \text{O:}& & 2s_2& =2s_3+s_4\end{array}}$

All solutions to this system of linear equations are of the following form, where r is any real number:

${\displaystyle \begin{array}{rl}{s}_1& =r\\ s_2& =2r\\ s_3& =r\\ s_4& =2r\end{array}}$

The choice of r = 1 yields the preferred solution,

${\displaystyle \begin{array}{rl}{s}_1& =1\\ s_2& =2\\ s_3& =1\\ s_4& =2\end{array}}$

which corresponds to the balanced chemical equation:

${\displaystyle {\text{CH}}_4^{}+2{\text{O}}_2^{}⟶{\text{CO}}_2^{}+2{\text{H}}_2^{}\text{O}}$

Matrix method

The system of linear equations introduced in the previous section can also be written using an efficient matrix formalism. First, to unify the reactant and product stoichiometric coefficients s_j, let us introduce the quantity

${\displaystyle {\nu }_j=\{\begin{array}{ll}-s_j& \text{for a reactant}\\ +s_j& \text{for a product}\end{array}}$

called stoichiometric number, which simplifies the linear equations to

${\displaystyle \sum _{j=1}^J{a}_{ij}{\nu }_j=0}$

where J is the total number of reactant and product substances (formulas) in the chemical equation.

Placement of the values a_ij at row i and column j of the composition matrix

A =${\displaystyle \left[\begin{array}{cccc}{a}_{1,1}& a_{1,2}& \cdots & a_{1,J}\\ a_{2,1}& a_{2,2}& \cdots & a_{2,J}\\ ⋮& ⋮& \ddots & ⋮\end{array}\right]}$

and arrangement of the stoichiometric numbers into the stoichiometric vector

ν =${\displaystyle \left[\begin{array}{c}{\nu }_1\\ {\nu }_2\\ ⋮\\ {\nu }_J\end{array}\right]}$

allows the system of equations to be expressed as a single matrix equation:

Aν = 0

Like previously, any nonzero stoichiometric vector ν, which solves the matrix equation, will balance the chemical equation.

The set of solutions to the matrix equation is a linear space called the kernel of the matrix A. For this space to contain nonzero vectors ν, i.e. to have a positive dimension J_N, the columns of the composition matrix A must not be linearly independent. The problem of balancing a chemical equation then becomes the problem of determining the J_N-dimensional kernel of the composition matrix. It is important to note that only for J_N = 1 will there be a unique preferred solution to the balancing problem. For J_N > 1 there will be an infinite number of preferred solutions with J_N of them linearly independent. If J_N = 0, there will be only the unusable trivial solution, the zero vector.

Techniques have been developed to quickly calculate a set of J_N independent solutions to the balancing problem, which are superior to the inspection and algebraic method in that they are determinative and yield all solutions to the balancing problem.

Example

Using the same chemical equation again, write the corresponding matrix equation:

${\displaystyle {\mathit{\text{s}}}_1^{}{\text{CH}}_4^{}+{\mathit{\text{s}}}_2^{}{\text{O}}_2^{}⟶{\mathit{\text{s}}}_3^{}{\text{CO}}_2^{}+{\mathit{\text{s}}}_4^{}{\text{H}}_2^{}\text{O}}$

${\displaystyle \begin{array}{c}\text{C:}\\ \text{H:}\\ \text{O:}\end{array}\left[\begin{array}{cccc}1& 0& 1& 0\\ 4& 0& 0& 2\\ 0& 2& 2& 1\end{array}\right]\left[\begin{array}{c}{\nu }_1\\ {\nu }_2\\ {\nu }_3\\ {\nu }_4\end{array}\right]=\mathbf{0}}$

Its solutions are of the following form, where r is any real number:

${\displaystyle \left[\begin{array}{c}{\nu }_1\\ {\nu }_2\\ {\nu }_3\\ {\nu }_4\end{array}\right]=\left[\begin{array}{c}-s_1\\ -s_2\\ s_3\\ s_4\end{array}\right]=r\left[\begin{array}{c}-1\\ -2\\ 1\\ 2\end{array}\right]}$

The choice of r = 1 and a sign-flip of the first two rows yields the preferred solution to the balancing problem:

${\displaystyle \left[\begin{array}{c}-{\nu }_1\\ -{\nu }_2\\ {\nu }_3\\ {\nu }_4\end{array}\right]=\left[\begin{array}{c}{s}_1\\ s_2\\ s_3\\ s_4\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 1\\ 2\end{array}\right]}$

Ionic equations

An ionic equation is a chemical equation in which electrolytes are written as dissociated ions. Ionic equations are used for single and double displacement reactions that occur in aqueous solutions.

For example, in the following precipitation reaction:

${\displaystyle {\text{CaCl}}_2^{}+2{\text{AgNO}}_3^{}⟶\text{Ca}{({\text{NO}}_3^{})}_2^{}+2\text{AgCl}\downarrow }$

the full ionic equation is:

${\displaystyle {\text{Ca}}^{2+}+2{\text{Cl}}^{-}+2{\text{Ag}}^{+}+2{\text{NO}}_3^{-}⟶{\text{Ca}}^{2+}+2{\text{NO}}_3^{-}+2\text{AgCl}\downarrow }$

or, with all physical states included:

${\displaystyle {\text{Ca}}^{2+}(\text{aq})+2{\text{Cl}}^{-}(\text{aq})+2{\text{Ag}}^{+}(\text{aq})+2{\text{NO}}_3^{-}(\text{aq})⟶{\text{Ca}}^{2+}(\text{aq})+2{\text{NO}}_3^{-}(\text{aq})+2\text{AgCl}\downarrow }$

In this reaction, the Ca²⁺ and the NO₃⁻ ions remain in solution and are not part of the reaction. That is, these ions are identical on both the reactant and product side of the chemical equation. Because such ions do not participate in the reaction, they are called spectator ions. A net ionic equation is the full ionic equation from which the spectator ions have been removed. The net ionic equation of the proceeding reactions is:

${\displaystyle 2{\text{Cl}}^{-}+2{\text{Ag}}^{+}⟶2\text{AgCl}\downarrow }$

or, in reduced balanced form,

${\displaystyle {\text{Ag}}^{+}+{\text{Cl}}^{-}⟶\text{AgCl}\downarrow }$

In a neutralization or acid/base reaction, the net ionic equation will usually be:

${\displaystyle {\text{H}}^{+}(\text{aq})+{\text{OH}}^{-}(\text{aq})⟶{\text{H}}_2^{}\text{O}(\text{l})}$

There are a few acid/base reactions that produce a precipitate in addition to the water molecule shown above. An example is the reaction of barium hydroxide with phosphoric acid, which produces not only water but also the insoluble salt barium phosphate. In this reaction, there are no spectator ions, so the net ionic equation is the same as the full ionic equation.

${\displaystyle 3\text{Ba}{(\text{OH})}_2^{}+2{\text{H}}_3^{}{\text{PO}}_4^{}⟶6{\text{H}}_2^{}\text{O}+{\text{Ba}}_3^{}{({\text{PO}}_4^{})}_2^{}\downarrow }$

${\displaystyle 3{\text{Ba}}^{2+}+6{\text{OH}}^{-}+6{\text{H}}^{+}+\underset{\text{phosphate}}{\underbrace{2{\text{PO}}_4^{3-}}}⟶6{\text{H}}_2^{}\text{O}+\underset{\text{barium}\text{phosphate}}{\overset{}{\underbrace{{\text{Ba}}_3^{}{({\text{PO}}_4^{})}_2^{}\downarrow }}}}$

Double displacement reactions that feature a carbonate reacting with an acid have the net ionic equation:

${\displaystyle 2{\text{H}}^{+}+\underset{\text{carbonate}}{\underbrace{{\text{CO}}_3^{2-}}}⟶{\text{H}}_2^{}\text{O}+{\text{CO}}_2^{}\uparrow }$

If every ion is a "spectator ion" then there was no reaction, and the net ionic equation is null.

Generally, if z_j is the multiple of elementary charge on the j-th molecule, charge neutrality may be written as:

${\displaystyle \sum _{j=1}^J{z}_j{\nu }_j=0}$

where the ν_j are the stoichiometric coefficients described above. The z_j may be incorporated as an additional row in the a_ij matrix described above, and a properly balanced ionic equation will then also obey:

${\displaystyle \sum _{j=1}^J{a}_{ij}{\nu }_j=0}$