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Affine transformation - Wikipedia
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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Parent (intermediate) annotation
Open itIn 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
Original toplevel document
Affine transformation - Wikipedias 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
Flashcard 1729653247244
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Parent (intermediate) annotation
Open itIn 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.
Original toplevel document
Affine transformation - Wikipedias 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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Parent (intermediate) annotation
Open itmap [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>
Original toplevel document
Affine transformation - Wikipedias 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
Flashcard 1729656392972
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Parent (intermediate) annotation
Open itAn 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.
Original toplevel document
Affine transformation - Wikipedias 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
Flashcard 1729657965836
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Parent (intermediate) annotation
Open itAn 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.
Original toplevel document
Affine transformation - Wikipedias 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
Flashcard 1729659538700
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Parent (intermediate) annotation
Open itAn 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.
Original toplevel document
Affine transformation - Wikipedias 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
is characterised by the following properties:
has independent increments: for every 
the future increments 
, are independent of the past values 
, 
is normally distributed with mean 
and variance 
, 
, 
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Wiener process - Wikipedia
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
Flashcard 1731730476300
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Parent (intermediate) annotation
Open itThe 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 ,
Original toplevel document
Wiener process - WikipediaBrownian 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
Flashcard 1731732835596
| status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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Parent (intermediate) annotation
Open itThe 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
Original toplevel document
Wiener process - WikipediaBrownian 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
Flashcard 1731734408460
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Parent (intermediate) annotation
Open itThe 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 .
Original toplevel document
Wiener process - WikipediaBrownian 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
Flashcard 1731736767756
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Parent (intermediate) annotation
Open itad> 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>
Original toplevel document
Wiener process - WikipediaBrownian 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
Flashcard 1731738340620
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Parent (intermediate) annotation
Open itr 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>
Original toplevel document
Wiener process - WikipediaBrownian 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 Goodies
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
Flashcard 1737415331084
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Parent (intermediate) annotation
Open itGenerator expressions are similar to list comprehensions, but with parentheses instead of square brackets:
Original toplevel document
The Goodiesuse 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
Flashcard 1737416903948
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Parent (intermediate) annotation
Open itGenerator expressions are similar to list comprehensions, but with parentheses instead of square brackets:
Original toplevel document
The Goodiesuse 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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Hilbert space - Wikipedia
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 - Wikipedia
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: |
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Hilbert space - Wikipedia
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
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Hilbert space - Wikipedia
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 - Wikipedia
→ 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
Flashcard 1796944563468
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Parent (intermediate) annotation
Open itthe 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.
Original toplevel document
Hilbert space - Wikipediable 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
Flashcard 1798939479308
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Parent (intermediate) annotation
Open itAn operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.
Original toplevel document
Hilbert space - Wikipediax · 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
Flashcard 1798941052172
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Parent (intermediate) annotation
Open itAn operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.
Original toplevel document
Hilbert space - Wikipediax · 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
Flashcard 1798942625036
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|---|---|---|---|---|---|---|---|
| repetition number in this series | 0 | memorised on | scheduled repetition | ||||
| scheduled repetition interval | last repetition or drill |
Parent (intermediate) annotation
Open itAn operation on pairs of vectors, that satisfies the three properties of the dot product, is known as a (real) inner product.
Original toplevel document
Hilbert space - Wikipediax · 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
Flashcard 1798944984332
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Parent (intermediate) annotation
Open itThe 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
Original toplevel document
Hilbert space - Wikipediat 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
Flashcard 1798947343628
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Open itMultivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist.
Original toplevel document
Hilbert space - Wikipedia\|\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 ∞
Flashcard 1798949702924
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| repetition number in this series | 0 | memorised on | scheduled repetition | ||||
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Parent (intermediate) annotation
Open itMultivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist.
Original toplevel document
Hilbert space - Wikipedia\|\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 ∞
Flashcard 1798952062220
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| repetition number in this series | 0 | memorised on | scheduled repetition | ||||
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Open itthe completeness of Euclidean space means that a series that converges absolutely also converges in the ordinary sense.
Original toplevel document
Hilbert space - Wikipedia→ 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
Flashcard 1798954421516
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Parent (intermediate) annotation
Open itThe 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:
Original toplevel document
Hilbert space - Wikipediato 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: |
Flashcard 1798955994380
| 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 |
Parent (intermediate) annotation
Open itThe 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:
Original toplevel document
Hilbert space - Wikipediato 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: |
Flashcard 1798957567244
| status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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| repetition number in this series | 0 | memorised on | scheduled repetition | ||||
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Parent (intermediate) annotation
Open itThe 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:
Original toplevel document
Hilbert space - Wikipediato 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: |
Flashcard 1798959140108
| 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 |
Parent (intermediate) annotation
Open itThe 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:
Original toplevel document
Hilbert space - Wikipediato 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: |
Flashcard 1798964645132
| status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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Open itA 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
Original toplevel document
Hilbert space - Wikipediaw ¯ . {\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
Flashcard 1798967004428
| status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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Parent (intermediate) annotation
Open itA 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.
Original toplevel document
Hilbert space - Wikipediaw ¯ . {\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
Flashcard 1798969363724
| 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 |
Parent (intermediate) annotation
Open itA 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>
Original toplevel document
Hilbert space - Wikipediaw ¯ . {\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