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#topology

In geometry, an **affine transformation**, **affine map**^{[1]} or an **affinity** (from the Latin, *affinis*, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

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s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#topology

In geometry, an **affine transformation**, **affine map**^{[1]} or an **affinity** (from the Latin, *affinis*, "connected with") is a function between affine spaces which preserves points, straight lines and planes.

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In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, tho

s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

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#topology

Question

Answer

points, straight lines and planes

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes.

s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#topology

An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

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last reprioritisation on | reading queue position [%] | |||

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map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. <span>An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. <span><body><html>

s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

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#topology

Question

affine transformation does not necessarily preserve [...] between lines

Answer

angles

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An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

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#topology

Question

affine transformation does not necessarily preserve [...] between points

Answer

distances

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

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#topology

Question

An affine transformation preserve [...] between points lying on a straight line.

Answer

ratios of distances

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

s related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation. <span>In geometry, an affine transformation, affine map [1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination

#stochastics

The Wiener process * * is characterised by the following properties:^{[1]}

- a.s.
- has independent increments: for every the future increments , are independent of the past values ,
- has Gaussian increments: is normally distributed with mean and variance ,
- has continuous paths: With probability , is continuous in

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Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

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The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance ,

Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

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#stochastics

Question

The Wiener process * * has **[...]** increments: for every the future increments , are independent of the past values ,

Answer

independent

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The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance , has continuous paths: With

Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

Tags

#stochastics

Question

The Wiener process * * has independent increments: for every [...

Answer

the future increments , are independent of the past values ,

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scheduled repetition interval | last repetition or drill |

The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance , has continuous paths: With probability , is continuous in .

Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

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#stochastics

Question

The Wiener process has [...]: is normally distributed with mean and variance ,

Answer

Gaussian increments

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ad> The Wiener process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with mean and variance , has continuous paths: With probability , is continuous in . <html>

Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

Tags

#stochastics

Question

in the Wiener process * , * is normally distributed with [...

Answer

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r process is characterised by the following properties: [1] a.s. has independent increments: for every the future increments , are independent of the past values , has Gaussian increments: is normally distributed with <span>mean and variance , has continuous paths: With probability , is continuous in . <span><body><html>

Brownian motion 4.3 Time change 4.4 Change of measure 4.5 Complex-valued Wiener process 4.5.1 Self-similarity 4.5.2 Time change 5 See also 6 Notes 7 References 8 External links Characterisations of the Wiener process[edit source] <span>The Wiener process W t {\displaystyle W_{t}} is characterised by the following properties: [1] W 0 = 0 {\displaystyle W_{0}=0} a.s. W {\displaystyle W} has independent increments: for every t > 0 , {\displaystyle t>0,} the future increments W t + u − W t , {\displaystyle W_{t+u}-W_{t},} u ≥ 0 , {\displaystyle u\geq 0,} , are independent of the past values W s {\displaystyle W_{s}} , s ≤ t . {\displaystyle s\leq t.} W {\displaystyle W} has Gaussian increments: W t + u − W t {\displaystyle W_{t+u}-W_{t}} is normally distributed with mean 0 {\displaystyle 0} and variance u {\displaystyle u} , W t + u − W t ∼ N ( 0 , u ) . {\displaystyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} W {\displaystyle W} has continuous paths: With probability 1 {\displaystyle 1} , W t {\displaystyle W_{t}} is continuous in t {\displaystyle t} . The independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 then W t 1 −W s 1 and W t 2 −W s 2 are independent random variables, and the similar condition holds for

#python

Generator expressions are similar to list comprehensions, but
with parentheses instead of square brackets:

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use you can’t put a print statement inside the loop. I suggest that you use them only if the computation is simple enough that you are likely to get it right the first time. And for beginners that means never. 19.3 Generator expressions <span>Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets: >>> g = (x**2 for x in range(5)) >>> g at 0x7f4c45a786c0> The result is a generator object that knows how to iterate through a sequence of values. But unlike a

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#python

Question

[...] are similar to list comprehensions, but
with parentheses instead of square brackets:

Answer

Generator expressions

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Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets:

use you can’t put a print statement inside the loop. I suggest that you use them only if the computation is simple enough that you are likely to get it right the first time. And for beginners that means never. 19.3 Generator expressions <span>Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets: >>> g = (x**2 for x in range(5)) >>> g at 0x7f4c45a786c0> The result is a generator object that knows how to iterate through a sequence of values. But unlike a

Tags

#python

Question

Generator expressions are similar to [...], but
with parentheses instead of square brackets:

Answer

list comprehensions

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scheduled repetition interval | last repetition or drill |

Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets:

use you can’t put a print statement inside the loop. I suggest that you use them only if the computation is simple enough that you are likely to get it right the first time. And for beginners that means never. 19.3 Generator expressions <span>Generator expressions are similar to list comprehensions, but with parentheses instead of square brackets: >>> g = (x**2 for x in range(5)) >>> g at 0x7f4c45a786c0> The result is a generator object that knows how to iterate through a sequence of values. But unlike a

#hilbert-space

An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.

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x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of t

#hilbert-space

The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | *z* | which is defined as the square root of the product of z with its complex conjugate:

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to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

#hilbert-space

A second example is the space ℂ^{2} whose elements are pairs of complex numbers *z* = (*z*_{1}, *z*_{2}) . Then the inner product of z with another such vector *w* = (*w*_{1},*w*_{2}) is given by

The real part of 〈*z*,*w*〉 is then the four-dimensional Euclidean dot product.

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w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This is complex-valued. The real part of ⟨z,w⟩ gives the usual two-dimensional Euclidean dot product. <span>A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ). Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by ⟨ z , w ⟩ = z 1 w ¯ 1 + z 2 w ¯ 2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z

#hilbert-space

The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted || **x** || , and to the angle θ between two vectors **x** and **y** by means of the formula

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t product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. <span>The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\|\mathbf {x} \|\,\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacem

#hilbert-space

the *completeness* of Euclidean space means that a series that converges absolutely also converges in the ordinary sense.

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→ 0 as N → ∞ . {\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.} <span>This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square

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the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product.

ble spaces 9 Orthogonal complements and projections 10 Spectral theory 11 See also 12 Remarks 13 Notes 14 References 15 External links Definition and illustration[edit source] Motivating example: Euclidean space[edit source] One of <span>the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ 3 , and equipped with the dot product. The dot product takes two vectors x and y, and produces a real number x · y. If x and y are represented in Cartesian coordinates, then the dot product is defined by

Tags

#hilbert-space

Question

An operation on pairs of vectors, that satisfies [...], is known as a (real) inner product.

Answer

the three properties of the dot product

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An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.

x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of t

Tags

#hilbert-space

Question

An operation on [...], that satisfies the three properties of the dot product, is known as a (real) inner product.

Answer

pairs of vectors

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.

x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of t

Tags

#hilbert-space

Question

An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a [...]

Answer

(real) inner product.

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An operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.

x · y = y · x. It is linear in its first argument: (ax 1 + bx 2 ) · y = ax 1 · y + bx 2 · y for any scalars a, b, and vectors x 1 , x 2 , and y. It is positive definite: for all vectors x, x · x ≥ 0 , with equality if and only if x = 0. <span>An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of t

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#hilbert-space

Question

The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both [...] and **[...]**

Answer

length and angle

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The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted || x || , and to the angle θ between two vectors x and y by means of the formula

t product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. <span>The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\|\mathbf {x} \|\,\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacem

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#hilbert-space

Question

Answer

Multivariable calculus in Euclidean space

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Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist.

\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in orange). <span>Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞

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#hilbert-space

Question

Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that [...]

Answer

limits exist.

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist.

\|\mathbf {y} \|\,\cos \theta \,.} [imagelink] Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in orange). <span>Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞

Tags

#hilbert-space

Question

the *[...]* of Euclidean space means that a series that converges absolutely also converges in the ordinary sense.

Answer

completeness

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the completeness of Euclidean space means that a series that converges absolutely also converges in the ordinary sense.

→ 0 as N → ∞ . {\displaystyle \left\|\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}\right\|\to 0\quad {\text{as }}N\to \infty \,.} <span>This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square

Tags

#hilbert-space

Question

The complex plane ℂ is equipped with a notion of [...]

Answer

magnitude

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The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

Tags

#hilbert-space

Question

The notion of magnitude in the complex plane is [...]

Answer

the complex modulus

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scheduled repetition interval | last repetition or drill |

The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

Tags

#hilbert-space

Question

the complex modulus is defined as [...description...]

Answer

the square root of the product of z with its complex conjugate

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus | z | which is defined as the square root of the product of z with its complex conjugate:

to 0\quad {\text{as }}N\to \infty \,.} This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers. <span>The complex plane denoted by ℂ is equipped with a notion of magnitude, the complex modulus |z| which is defined as the square root of the product of z with its complex conjugate: | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: |

Tags

#hilbert-space

Question

the elements of space ℂ^{2} are [...]

Answer

pairs of complex numbers

*z* = (*z*_{1}, *z*_{2})

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A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ) . Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by The real part of 〈z,w〉 is then the four-dimensional Euclidean dot product. </sp

w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This is complex-valued. The real part of ⟨z,w⟩ gives the usual two-dimensional Euclidean dot product. <span>A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ). Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by ⟨ z , w ⟩ = z 1 w ¯ 1 + z 2 w ¯ 2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z

Tags

#hilbert-space

Question

the inner product of two vectors (z and *w) *in the space ℂ^{2 }is given by [...]

Answer

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ) . Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by The real part of 〈z,w〉 is then the four-dimensional Euclidean dot product.

w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This is complex-valued. The real part of ⟨z,w⟩ gives the usual two-dimensional Euclidean dot product. <span>A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ). Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by ⟨ z , w ⟩ = z 1 w ¯ 1 + z 2 w ¯ 2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z

Tags

#hilbert-space

Question

The real part of the inner product〈*z*,*w*〉in the space ℂ^{2} is the [...] in the Euclidean space

Answer

four-dimensional dot product

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ) . Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by The real part of 〈z,w〉 is then the <span>four-dimensional Euclidean dot product. <span><body><html>

w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This is complex-valued. The real part of ⟨z,w⟩ gives the usual two-dimensional Euclidean dot product. <span>A second example is the space ℂ 2 whose elements are pairs of complex numbers z = (z 1 , z 2 ). Then the inner product of z with another such vector w = (w 1 ,w 2 ) is given by ⟨ z , w ⟩ = z 1 w ¯ 1 + z 2 w ¯ 2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z