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Encyclopedia > Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). Note that scalar multiplication is different than scalar product which is an inner product between two vectors. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ... // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...


More specifically, if K is a field and V is a vector space over K, then scalar multiplication is a function from K × V to V. The result of applying this function to c in K and v in V is cv. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a function returns a unique output for a given input. ...


Scalar multiplication obeys the following rules (vector in boldface): Bold Bold, see Bold (disambiguation). ...

  • Left distributivity: (c + d)v = cv + dv;
  • Right distributivity: c(v + w) = cv + cw;
  • Associativity: (cd)v = c(dv);
  • Multiplying by 1 does not change a vector: 1v = v;
  • Multiplying by 0 gives the null vector: 0v = 0;
  • Multiplying by -1 gives the additive inverse: (-1)v = -v.

Here + is addition either in the field or in the vector space, as appropriate; and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field. In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, associativity is a property that a binary operation can have. ... In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... 3 + 2 with apples Addition is the most basic operation of arithmetic. ... In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ...


Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space. A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector. In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a symmetry group describes all symmetries of objects. ... Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ...


As a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field. When V is Kn, then scalar multiplication is defined component-wise.


The same idea goes through with no change if K is a commutative ring and V is a module over K. K can even be a rig, but then there is no additive inverse. If K is not commutative, then the only change is that the order of the multiplication may be reversed from what we've written above. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...


See also


  Results from FactBites:
 
PowerPedia:Scalar field theory - PESWiki (4720 words)
Scalar waves are hypothetical waves, which differ from the conventional electromagnetic transverse waves by one oscillation level parallel to the direction of propagation; they thus have characteristics of longitudinal waves.
Scalar multiplication may be viewed as an external binary operation or as an action of the field on the vector space.
A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector.
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