...pace in which the action of addition and the action of multiplication by a scalar are continuous with respect to the given topology in $L$. See [[Topological
279 bytes (45 words) - 20:50, 11 April 2014
...r space with respect to the operations of addition and multiplication by a scalar defined in $E$. A set $L+x_0$, where $x_0\in E$, is called a [[Linear varie
349 bytes (61 words) - 22:17, 30 November 2018
...analysis|vector analysis]] are vectors which are functions of one or more scalar arguments.
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...r|Vector]]). These include linear operations, viz. addition of vectors and multiplication of a vector by a number.
The operation of multiplication of a vector by a number has the properties:
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The application of the Hamilton operator to a scalar function $ f $,
which is understood as multiplication of the "vector" $ \nabla $
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...ear space, i.e. a vector space with a metric such that addition and scalar multiplication are continuous, then there is an invariant metric $\rho'$ on $X$ that is eq
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...the closure of the vector sum. Addition, subtraction, multiplication by a scalar, and a topology are introduced in the space of convex sets, that space beco
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...als with the operations of addition of screws, multiplication by a number, scalar and vector products, etc. In this context, the operations of helical calcul
For instance, the scalar product of two screws is equal to the product of their complex moduli and t
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The convolution has the basic properties of multiplication, namely,
...nd of multiplication by a scalar, with the operation of convolution as the multiplication of elements, and with the norm
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while multiplication by a number $ \lambda \in \mathbf R $
and is a Euclidean space with respect to the scalar product
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...ion $A$ of a Euclidean space preserving the lengths (or, equivalently, the scalar product) of vectors. Orthogonal transformations and only they can transfer
...orthogonal transformations in a Euclidean space is a group with respect to multiplication of transformations — the [[Orthogonal group|orthogonal group]] of the giv
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...more dimensions. However, if one drops the requirement of commutativity of multiplication, then it is possible to construct a number system from the points of $ 4
"basic units" ) and the following multiplication table of the "basic units" :
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...contravariant tensors of the same valency and
multiplication of them by a
scalar. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.
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now defines a multiplication operation on the direct sum of linear spaces $ A _ {1} = A \oplus A $,
multiplication table
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A tensor considered up to multiplication by an arbitrary function (cf. [[Tensor on a vector space|Tensor on a vector
is a scalar-valued function. Most frequently (in applications), the function $ \tau $
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...h are invariant with respect to a shift, become (under certain conditions) multiplication operators in the image space. In particular, the convolution of two functio
and differentiation induces multiplication by the independent variable:
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is homogeneous with respect to multiplication by positive numbers and satisfies the condition $ f ( xx ^ {*} ) = f ( x
with respect to the scalar product defined by the form $ s $.
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of exterior multiplication by $ \Omega $;
of interior multiplication by $ \Omega $;
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as subspaces, and with a multiplication governed by the rule
their (geometric) product splits into a scalar and a so-called bivector part:
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...in $\mathbf{E} ^ { n }$. $\mathcal{K} ^ { n }$ with Minkowski addition and multiplication by non-negative scalars is a convex cone. The mapping $K \mapsto h _ { K }$
($\langle \, .\, ,\, . \, \rangle$ being the scalar product) is the support function, maps this cone isomorphically into the sp
4 KB (596 words) - 15:30, 1 July 2020