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Flashcard 1729703841036

Tags
#probability
Question
The negative binomial distribution also arises as a [...] of Poisson distributions
Answer
continuous mixture

Can be used to model over dispersed count observations, known as Gamma-Poisson distribution.

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The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution.

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Negative binomial distribution - Wikipedia
) . {\displaystyle \operatorname {Poisson} (\lambda )=\lim _{r\to \infty }\operatorname {NB} \left(r,{\frac {\lambda }{\lambda +r}}\right).} Gamma–Poisson mixture[edit source] <span>The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution. That is, we can view the negative binomial as a Poisson(λ) distribution, where λ is itself a random variable, distributed as a gamma distribution with shape = r and scale θ = p/(1 − p) or correspondingly rate β = (1 − p)/p. To display the intuition behind this statement, consider two independent Poisson processes, “Success” and “Failure”, with intensities p and 1 − p. Together, the Success and Failure pr







#fourier-analysis

When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:

The functions f and g are orthogonal when this integral is zero: As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space.

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Orthogonal functions - Wikipedia
ia (Redirected from Orthogonal function) Jump to: navigation, search In mathematics, orthogonal functions belong to a function space which is a vector space (usually over R) that has a bilinear form. <span>When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: ⟨ f , g ⟩ = ∫ f ( x ) g ( x ) d x . {\displaystyle \langle f,g\rangle =\int f(x)g(x)\,dx.} The functions f and g are orthogonal when this integral is zero: ⟨ f , g ⟩ = 0. {\displaystyle \langle f,\ g\rangle =0.} As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Suppose {f n }, n = 0, 1, 2, … is a sequence of orthogonal functions. If f n has positive support then ⟨ f n




Flashcard 1732605512972

Tags
#linear-algebra #matrix-decomposition
Question

If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is [...] and its inverse is given by [...]

Answer
nonsingular

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Eigendecomposition of a matrix - Wikipedia
amp;0\\1&3\\\end{bmatrix}}{\begin{bmatrix}-2c&0\\c&d\\\end{bmatrix}}={\begin{bmatrix}1&0\\0&3\\\end{bmatrix}},[c,d]\in \mathbb {R} } Matrix inverse via eigendecomposition[edit source] Main article: Inverse matrix <span>If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is nonsingular and its inverse is given by A − 1 = Q Λ − 1 Q − 1 {\displaystyle \mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1}} Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate: [ Λ







Flashcard 1732608658700

Tags
#linear-algebra #matrix-decomposition
Question

[...] has real eigenvalues and the eigenvectors can be chosen to be orthogonal

Answer

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Eigendecomposition of a matrix - Wikipedia
{\displaystyle A=A^{*}} ), which implies that it is also complex normal, the diagonal matrix Λ has only real values, and if A is unitary, Λ takes all its values on the complex unit circle. Real symmetric matrices[edit source] <span>As a special case, for every N×N real symmetric matrix, the eigenvalues are real and the eigenvectors can be chosen such that they are orthogonal to each other. Thus a real symmetric matrix A can be decomposed as A = Q Λ Q T {\displaystyle \mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T}} where Q is an orthogonal matrix, and Λ is a diagonal matrix whose entries are the eigenvalues of A. Useful facts[edit source] Useful facts regarding eigenvalues[edit source] The product of the eigenvalues is equal to the determinant of A det







#distributions
The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").
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Compound probability distribution - Wikipedia
istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als




Flashcard 1732623338764

Tags
#distributions
Question
The compound distribution is the result of [...] the latent random variables representing the parameters of the parametrized distribution
Answer
marginalizing out

Also called integrating over

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The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").

Original toplevel document

Compound probability distribution - Wikipedia
istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als







Flashcard 1732624911628

Tags
#distributions
Question
The compound distribution is integrated over [...] that represents the parameters of the parametrized distribution
Answer
latent random variables

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The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").

Original toplevel document

Compound probability distribution - Wikipedia
istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als







#mathematical-structures

In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

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Morphism - Wikipedia
st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth




#mathematical-structures
In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.
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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphism

Original toplevel document

Morphism - Wikipedia
st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth




Flashcard 1732635921676

Tags
#mathematical-structures
Question
[...] refers to a structure-preserving map from one mathematical structure to another.
Answer
morphism

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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.

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Morphism - Wikipedia
st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth







Flashcard 1732637494540

Tags
#mathematical-structures
Question
morphism refers to a [...] from one mathematical structure to another.
Answer
structure-preserving map

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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.

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Morphism - Wikipedia
st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth







In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself.
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Endomorphism - Wikipedia
his article is about the mathematical concept. For other uses, see Endomorphic. [imagelink] Orthogonal projection onto a line, m, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism. <span>In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map, f: V → V, and an endomorphism of a group, G, is a group homomorphism f: G → G. In general, we can talk about endomorphi




Flashcard 1732642475276

Question
In mathematics, an [...] is a morphism (or homomorphism) from a mathematical object to itself.
Answer
endomorphism

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In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself.

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Endomorphism - Wikipedia
his article is about the mathematical concept. For other uses, see Endomorphic. [imagelink] Orthogonal projection onto a line, m, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism. <span>In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map, f: V → V, and an endomorphism of a group, G, is a group homomorphism f: G → G. In general, we can talk about endomorphi







In operator theory, a multiplication operator is an operator Tf defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f . That is,

for all φ in the domain of Tf , and all x in the domain of φ (which is the same as the domain of f ).

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Multiplication operator - Wikipedia
This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) (Learn how and when to remove this template message) <span>In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, T f φ ( x ) = f ( x ) φ ( x ) {\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of th




Multiplication operators generalize the notion of operator given by a diagonal matrix.
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Multiplication operator - Wikipedia
{\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. <span>Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of the results of operator theory is a spectral theorem, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication




Flashcard 1732653223180

Question
[...] generalize the notion of operator given by a diagonal matrix.
Answer
Multiplication operators

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Multiplication operators generalize the notion of operator given by a diagonal matrix.

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Multiplication operator - Wikipedia
{\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. <span>Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of the results of operator theory is a spectral theorem, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication







Flashcard 1732655058188

Question
Multiplication operators generalize the notion of operator given by a [...].

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Multiplication operators generalize the notion of operator given by a diagonal matrix.

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Multiplication operator - Wikipedia
{\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. <span>Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of the results of operator theory is a spectral theorem, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication







Flashcard 1732657679628

Question

In operator theory, the value of a [...] at a function φ is given by multiplication by a fixed function f .

Answer
multiplication operator

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In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f . That is, for all φ in the d

Original toplevel document

Multiplication operator - Wikipedia
This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) (Learn how and when to remove this template message) <span>In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, T f φ ( x ) = f ( x ) φ ( x ) {\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of th







Flashcard 1732659252492

Question

In operator theory, the value of a multiplication operator at a function φ is given by [...] .

Answer
multiplication by a fixed function f

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In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f . That is, for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f ).

Original toplevel document

Multiplication operator - Wikipedia
This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) (Learn how and when to remove this template message) <span>In operator theory, a multiplication operator is an operator T f defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, T f φ ( x ) = f ( x ) φ ( x ) {\displaystyle T_{f}\varphi (x)=f(x)\varphi (x)\quad } for all φ in the domain of T f , and all x in the domain of φ (which is the same as the domain of f). This type of operators is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of th







#matrices #spectral-theorem
In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, , which are as simple as one can hope to find.
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Spectral theorem - Wikipedia
putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per




Flashcard 1732662398220

Tags
#matrices #spectral-theorem
Question
In general, the spectral theorem identifies a class of [...] that can be modeled by multiplication operators
Answer
linear operators

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In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators

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Spectral theorem - Wikipedia
putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per







Flashcard 1732663971084

Tags
#matrices #spectral-theorem
Question
In general, the spectral theorem identifies a class of linear operators that can be modeled by [...]

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In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators

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Spectral theorem - Wikipedia
putations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. <span>In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical per







ok, si es de ves en cuando que lo realizas esta bien, porque todos los dias es malo..
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What is the meaning of &quot;de ves en cuando&quot;? | SpanishDict Answers
uando"? [imagelink] 2 Votes YBANEZ Please explain to me what my good friend want to convey when she said, "a <span>ok, si es de ves en cuando que lo realizas esta bien, porque todos los dias es malo.." Thank you very much Posted Mar 11, 2010 | 22310 views | link




Flashcard 1732697787660

Question
ok, si es [...] que lo realizas esta bien, porque todos los dias es malo..
Answer
de vez en cuando

de vez en vez o de cuando en cuando son también aceptable..

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ok, si es de ves en cuando que lo realizas esta bien, porque todos los dias es malo..

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What is the meaning of &quot;de ves en cuando&quot;? | SpanishDict Answers
uando"? [imagelink] 2 Votes YBANEZ Please explain to me what my good friend want to convey when she said, "a <span>ok, si es de ves en cuando que lo realizas esta bien, porque todos los dias es malo.." Thank you very much Posted Mar 11, 2010 | 22310 views | link







Flashcard 1732704603404

Question
▸ ¿pero qué [...] ? ¿que haga yo tu trabajo?
are you trying to get me to do your work, or what?
Answer
pretendes

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In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.
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Domain of a function - Wikipedia
main (disambiguation). [imagelink] Illustration showing f, a function from the pink domain X to the blue codomain Y. The yellow oval inside Y is the image of f. Both the image and the codomain are sometimes called the range of f. <span>In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. [1] Conversely, the set of values the function takes on as output is termed the image of th




Flashcard 1732720069900

Question
the domain of definition of a function is [...] for which the function is defined.
Answer
the set of "input" values

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In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

Original toplevel document

Domain of a function - Wikipedia
main (disambiguation). [imagelink] Illustration showing f, a function from the pink domain X to the blue codomain Y. The yellow oval inside Y is the image of f. Both the image and the codomain are sometimes called the range of f. <span>In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. [1] Conversely, the set of values the function takes on as output is termed the image of th







#functional-analysis
n mathematics, a function space is a set of functions between two fixed sets.
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Function space - Wikipedia
tional · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective   Constructions   Restriction · Composition · λ · Inverse   Generalizations   Partial · Multivalued · Implicit v t e I<span>n mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a




Flashcard 1732726099212

Tags
#functional-analysis
Question
a [...] is a set of functions between two fixed sets.
Answer
function space

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n mathematics, a function space is a set of functions between two fixed sets.

Original toplevel document

Function space - Wikipedia
tional · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective   Constructions   Restriction · Composition · λ · Inverse   Generalizations   Partial · Multivalued · Implicit v t e I<span>n mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a







#inner-product-space #vector-space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Vector space - Wikipedia
company, see Vector Space Systems. [imagelink] 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. <span>A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of v




Flashcard 1732731866380

Tags
#inner-product-space #vector-space
Question
A [...] is a collection vectors that can be added together and multiplied ("scaled") by numbers.
Answer
vector space

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A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Vector space - Wikipedia
company, see Vector Space Systems. [imagelink] 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. <span>A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of v







#linear-algebra

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × VK , where K is the field of scalars. In other words, a bilinear form is a function B : V × VK that is linear in each argument separately:

  • B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v)
  • B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v)
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Bilinear form - Wikipedia
Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one




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#linear-algebra
Question

a bilinear form is [...descriptive]

Answer
linear in each argument separately

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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K , where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is <span>linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) <span><body><html>

Original toplevel document

Bilinear form - Wikipedia
Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one