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Topic: Scalar multiplication

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	PowerPedia:Scalar field theory - PESWiki
		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.
	www.peswiki.com /index.php/PowerPedia:Scalar_field_theory (4720 words)

	Scalar multiplication - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-07)
		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).
		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.
	www.wikipedia.org /wiki/Scalar_multiplication (332 words)

	Scalar - Wikipedia, the free encyclopedia
		In mathematics, physics, and computing, a scalar is a quantity usually characterized by a single numeric value or not involving the concept of direction.
		In mathematics, scalars are components of vector spaces (and modules), usually real numbers which can be multiplied into vectors by scalar multiplication, or produced from vectors by scalar product.
		In physics, a scalar is a simple physical quantity that does not change under a change of coordinate system; for example, speed (180 km/h) is a scalar, while velocity (180 km/h north) is a vector.
	en.wikipedia.org /wiki/Scalar (265 words)

	Scalar multiplication (Site not responding. Last check: 2007-10-07)
		In mathematics, scalar multiplication is one of the basicoperations defining a vector space in linear algebra (or more generally, a module in abstract algebra).
		Scalar multiplication may be viewed as an external binaryoperation or as an action of the field on the vector space.
		A geometric interpertation to scalar multiplication is a streching orshrinking of a vector.
	www.therfcc.org /scalar-multiplication-291987.html (273 words)

	Encyclopedia: Scalar multiplication
		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, scalar multiplication is one of the basic operations defining a vector space or module in linear algebra.
		An example of an external binary operation is scalar multiplication in linear algebra.
	www.nationmaster.com /encyclopedia/Scalar-multiplication (1141 words)

	Scalar Multiplication
		A scalar is any constant value used as a scale factor applied to a vector.
		Thus, multiplication of a vector by a scalar is done in the obvious way, which is to multiply each coordinate of the vector by the scalar.
		When the scalar magnitude is greater than one, it is often called a gain factor, and when it is less than one, an attenuation.
	ccrma.stanford.edu /~jos/mdft/Scalar_Multiplication.html (183 words)

	Method and apparatus for elliptic curve cryptography and recording medium therefore - Patent 6876745
		Further, for the scalar multiplication d(x, y), a random number k is generated upon transformation of the affine coordinates (x, y) to the projective coordinates for thereby effectuating the transformation (x, y).fwdarw.[kx, ky, k] or alternatively (x, y).fwdarw.[k.sup.2 x, k.sup.3 y, k].
		In the elliptic curve cryptography, a scalar multiplication (SB) arithmetic for a given point R is adopted for the data encryption, generation of a digital signature and the verification of the digital signature.
		It is presumed that a projective coordinate component X.sub.0 of the x-coordinate of a given point R and a scalar value m are inputted and that a projective coordinate component X.sub.m of the x-coordinate of a point corresponding to m-multiple of R is to be outputted.
	www.freepatentsonline.com /6876745.html (11061 words)

	Vector space Summary
		In this case, the scalar (1/2) was multiplied by the vector (60 mph west) and the result is the vector (30 mph west).
		Scalar multiplication by negative unity yields the additive inverse of the vector:
		Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right.
	www.bookrags.com /Vector_space (4366 words)

	Matrix multiplication - Encyclopedia, History, Geography and Biography
		It's also easy to see why the number of columns in the proportions matrix has to be the same as the number of rows in the vectors matrix: they have to represent the same number of vectors.
		This notion of multiplication is important because if A and B are interpreted as linear transformations (which is almost universally done), then the matrix product AB corresponds to the composition of the two linear transformations, with B being applied first.
		When the underlying ring is commutative, for example, the real or complex number field, the two multiplications are the same.
	www.arikah.net /encyclopedia/Matrix_multiplication (1165 words)

	3.2 Arithmetic
		For non-commutative operations, such as matrix multiplication, the modifying forms use right multiplication, so a = b is equivalent to a = a b.
		the “” operator on matrices is overloaded to mean matrix multiplication, not componentwise multiplication, and “/” is defined to be multiplication* by the matrix inverse of the second argument.
		indicate when scalar operands are permitted by giving forms of the operators or functions where one of the inputs has one element only.
	libsh.org /ref/online/onlinese13.html (814 words)

	Linear transformation - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-07)
		In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
		Given again the finite dimensional case, if bases have been chosen, then the linear maps' composition corresponds to the matrix multiplication, the addition of linear maps corresponds to the matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
		A linear transformation f : V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring).
	www.wikipedia.org /wiki/Linear_transformation (1112 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		A controller is configured to control the scalar multiplier, the first vector multiplier and the second vector multiplier, to overlap scalar multiplies using a selected digit of the multiplier and vector multiplies using a modulus and the multiplicand.
		and the second vector multiplier 130, to overlap scalar multiplies using a selected digit of the multiplier, and vector multiplies using a modulus and the multiplicand, to thereby allow latency of Montgomery multiplication to be reduced to the latency of a single scalar multiplication.
		A Montgomery exponentiator according to Claim 4 wherein the scalar multiplier is further configured to multiply the least significant digit of the multiplicand by the first selected digit of the multiplier and by one over a negative of a least significant digit of the modulus to produce the scalar multiplier output.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=01/93012.011206&ELEMENT_SET=DECL (9766 words)

	commutative law -- Encyclopædia Britannica
		While commutativity holds for many systems, there are exceptions—as in quaternions in which commutativity of multiplication is invalid.
		The basic rules, or axioms, for addition and multiplication are shown in the table, and a set that satisfies all 10 of these rules is called a field.
		A ring satisfying the commutative law of multiplication (axiom 8) is known as a commutative ring.
	www.britannica.com /eb/article-9024995 (900 words)

	Device for the execution of a scalar multiplication of vectors - Patent 4566077
		A device for executing a scalar multiplication of vectors is constructed in the form of an interferometric adder for residue numbers.
		The result of the scalar multiplication is derivable as a positionally notated number from the interference pattern or interference patterns produced after the radiation through the phase modulators.
		With the invention, a result of the scalar multiplication may be obtained as a positionally notated number from the interference pattern or interference patterns generated after radiation through the phase modulator means.
	www.freepatentsonline.com /4566077.html (5818 words)

	Vector Space
		It's by no means a group operation (except for the case when we look at the set R of real numbers as a real vector space) because in a group operations both operands must come from the same set.
		Multiplication by a scalar is required to satisfy three additional laws: for u,v
		It's important to understand that an n-tuple is only then is regarded as a vector when it's considered an element of a set where two operations (addition and multiplication by a scalar) are defined.
	www.cut-the-knot.com /do_you_know/mul_scal1.shtml (310 words)

	Vector product
		Recall that “bcos θ” is the scalar component of vector b along the direction of vector a and “a cos θ” is the scalar component of vector a along the direction of vector b.
		The scalar component of vector in figure (i) is obtained by drawing perpendicular from the tip of the vector, b, on the direction of vector, a.
		Similarly, the scalar component of vector in figure (i) is obtained by drawing perpendicular from the tip of the vector, b, on the direction of vector, a.
	cnx.org /content/m13603/latest (1966 words)

	Sympathetic Vibratory Physics - John W. Keely's Sacred Science. (Site not responding. Last check: 2007-10-07)
		In such cases, an alternative interpretation of the multiplication is possible, where instead of changing the scale one changes the vector.
		The reader may find the description of multiplication of a vector by a number confusing if the vector is also considered a sort of "number" albeit a complex or hypercomplex number.
		Then what we have been talking about is the multiplication of a vector by a scalar, that is, multiplication of a complex or hypercomplex number by a single "ordinary number." As far as vectors are concerned, scalar multiplication is a unary operation because it is carried out on one vector only.
	www.svpvril.com /svpnotes/SCALAR_5629.html (369 words)

	you get: scalar multiplication (Site not responding. Last check: 2007-10-07)
		Scalar multiplication refers to the multiplication of a vector by a constant s, producing a vector in the same (for s>0) or opposite (for s<0) direction but of different length.
		Scalar multiplication There are two types of multiplication for matrices: scalar multiplication and matrix multiplication.
		Scalar multiplication Next: Addition Up: Introduction Previous: General Scalar multiplication Multiplying a matrix by a scalar (i.e.
	www.digital-webcams.de /scalar_multiplication.html (270 words)

	Matrix Multiplication: How to Multiply Two Matrices Together
		Scalar in which a single number is multiplied with every entry of a matrix
		Multiplication of an entire matrix by another entire matrix For the rest of the page, matrix multiplication will refer to this second category.
		In the scalar variety, every entry is multiplied by a number, called a scalar.
	www.mathwarehouse.com /algebra/matrix/multiply-matrix.php (429 words)

	Multiplications of Matrices
		Scalar multiplication consists of multiplying each element of a matrix by a given scalar.
		We use the terms scalar and scalar multiplication because, in abstract algebra, we often have the need to consider more general scalars than real numbers.
		Although the same number of operations are needed whether we use one matrix multiplication or two, it is easier to keep track of all of our information when we use one matrix multiplication.
	ceee.rice.edu /Books/LA/mult/index.html (1869 words)

	[No title]
		A6: (s+t)A=[(s+t)aij]=[saij+taij]=[saij]+[taij]=sA+tA, by the definition of scalar* multiplication for matrices, since scalar multiplication is distributive over scalar addition.
		A7: (st)A=[(st)aij]=[s(taij)]=s[taij]=s(tA), by the definition of scalar multiplication for matrices, since scalar multiplication is associative.
		A8: 1A=[1aij]=[aij]=A, by the definition of scalar* multiplication for matrices, since 1 is the multiplicative identity for scalars.
	www.math.wustl.edu /~victor/classes/ma309/s04.txt (787 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Multiplication of Vectors (called the inner product of two vectors): Two vectors can be multiplied only when the number of columns of the first equals the number of rows of the second.
		Multiplication of a Matrix by a scalar: Multiplication of a matrix by a scalar involves multiplying each element of the matrix by the scalar.
		Multiplication of matrices: Two matrices can be multiplied together only when the number of columns of the first matrix is equal to the number of rows of the second matrix.
	www-unix.oit.umass.edu /~bioep740/old/2001spring/source/matrix-intro.doc (2280 words)

	The const operator - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-10-07)
		The difference is that some people think of multiplication as a scalar operation not extending to other dimensions, whereas others think of scalar multiplication as the equivalent of the dot-product in 1-dimensional space (which it actually rationally is).
		The projection of a scalar value, rationalized as a vector in 1-dimension, onto the number line is always itself, since the vector and the number line are always going to be colinear to begin with.
		So, while some people claim that multiplication of scalars and the dot-product are completely different, since multiplication of scalars yields you an object having the same type as the operands, others will acknowledge the fact that multiplication of scalars is just the dot-product of two vectors applied to vectors in 1-dimension.
	www.gamedev.net /community/forums/viewreply.asp?ID=1938334 (5235 words)

	Relating Scalar and Matrix Multiplication
		We now have two sorts of multiplication that can be used on a matrix A; we can multiply it by a number or by another matrix.
		This sort of multiplication is given a special name; it is scalar multiplication, because the numbers are sometimes called scalars.
		In other words, multiplication by a scalar is the same as matrix multiplication by a diagonal matrix (of the right size) with the scalar on the diagonal.
	www.maths.abdn.ac.uk /~igc/tch/eg1006/notes/node115.html (151 words)

	Ma 305, Homework 4 Solutions (Site not responding. Last check: 2007-10-07)
		Scalar multiplication distributes over the sum of two vectors, and the sum of two scalars distribute over a vector, for
		to the scalar mult., an identity scalar (i.e.
		As in (a) above, the space is closed under the defined vector addition and scalar multiplication.
	www.ncsu.edu:8010 /Project25/MA305/Spring98/Homework/hw4_sol.html (633 words)

	Matrices
		For matrices with real number entries, real numbers are referred to as scalars.
		General multiplication In general we multiply two matrices by multiplying each row of the left matrix with each column of the right matrix.
		Matrices of appropriate sizes can be added, subtracted and multiplied with each other as well as being multiplied by scalars.
	science.kennesaw.edu /~nzumoff/3260F2004/Matrices.htm (612 words)

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