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| :''This article is about the transpose of a matrix. For other uses, see [[Transposition (disambiguation)|Transposition]]''
| | It is time to address the slow computer issues whether or not we do not recognize how. Just because your computer is working thus slow or keeps freezing up; refuses to imply to not address the problem plus fix it. You may or could not be aware that any computer owner must know that there are certain points which your computer requires to maintain the greatest performance. The sad truth is the fact that numerous individuals whom own a system have no idea which it needs routine maintenance really like their vehicles.<br><br>Before actually getting the software it really is best to check on the companies that make the software. If you may discover details found on the type of standing each firm has, perhaps the risk of malicious programs will be reduced. Software from reputed companies have helped me, and various other consumers, to make my PC run faster.. If the product description does not look good to you, does not include details about the software, refuses to include the scan functions, we should go for another 1 which ensures you're paying for what we wish.<br><br>If you compare registry cleaners you require a fast acting registry cleaning. It's no superior spending hours and your PC waiting for a registry cleaning to complete its task. We wish the cleaner to complete its task inside minutes.<br><br>Paid registry cleaners on the other hand, I have found, are usually inexpensive. They supply usual, free updates or at least cheap changes. This follows considering the software manufacturer requires to confirm their product is most effective in staying ahead of its competitors.<br><br>Another common cause of PC slow down is a corrupt registry. The registry is a surprisingly important component of computers running on Windows platform. When this gets corrupted a PC will slowdown, or worse, not start at all. Fixing the registry is easy with all the use of a system and [http://bestregistrycleanerfix.com/registry-reviver registry reviver].<br><br>The first thing we should do is to reinstall any program which shows the error. It's typical for many computers to have certain programs which need this DLL to show the error whenever you try plus load it up. If you see a particular program show the error, you must first uninstall which system, restart your PC and then resinstall the system again. This should substitute the damaged ac1st16.dll file plus remedy the error.<br><br>The disk requires room in purchase to run smoothly. By freeing up certain area from your disk, you are able to speed up the PC a bit. Delete all file in the temporary internet files folder, recycle bin, obvious shortcuts plus icons from the desktop which we do not utilize plus remove programs we do not employ.<br><br>Most folks make the mistake of trying to fix Windows registry by hand. I strongly recommend you don't do it. Unless you're a computer expert, I bet you'll invest hours plus hours learning the registry itself, let alone fixing it. And why if you waste a valuable time in learning plus fixing anything you understand nothing about? Why not let a smart and specialist registry cleaner do it for we? These software programs would be able to do the job inside a far better technique! Registry cleaners are quite affordable because well; you pay a 1 time fee and utilize it forever. Also, most professional registry cleaners are truly reliable plus effortless to use. If you require more info on how to fix Windows registry, just visit my website by clicking the link below! |
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| :''Note that this article assumes that matrices are taken over a commutative ring. These results may not hold in the non-commutative case.''
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| [[File:Matrix transpose.gif|thumb|200px|right|The transpose '''A'''<sup>T</sup> of a matrix '''A''' can be obtained by reflecting the elements along its main diagonal. Repeating the process on the transposed matrix returns the elements to their original position.]]
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| In [[linear algebra]], the '''transpose''' of a [[matrix (mathematics)|matrix]] '''A''' is another matrix '''A'''<sup>T</sup> (also written '''A'''′, '''A'''<sup>tr</sup>,<sup>t</sup>'''A''' or '''A'''<sup>t</sup>) created by any one of the following equivalent actions:
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| * reflect '''A''' over its [[main diagonal]] (which runs from top-left to bottom-right) to obtain '''A'''<sup>T</sup>
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| * write the rows of '''A''' as the columns of '''A'''<sup>T</sup>
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| * write the columns of '''A''' as the rows of '''A'''<sup>T</sup>
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| Formally, the ''i'' th row, ''j'' th column element of '''A'''<sup>T</sup> is the ''j'' th row, ''i'' th column element of '''A''':
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| :<math>[\mathbf{A}^\mathrm{T}]_{ij} = [\mathbf{A}]_{ji}</math>
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| If '''A''' is an {{nowrap|''m'' × ''n''}} matrix then '''A'''<sup>T</sup> is an {{nowrap|''n'' × ''m''}} matrix. | |
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| The transpose of a matrix was introduced in 1858 by the British mathematician [[Arthur Cayley]].<ref>Arthur Cayley (1858) [http://books.google.com/books?id=flFFAAAAcAAJ&pg=PA31#v=onepage&q&f=false "A memoir on the theory of matrices,"] ''Philosophical Transactions of the Royal Society of London'', '''148''' : 17-37. The transpose (or "transposition") is defined on page 31.</ref>
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| == Examples ==
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| *<math>\begin{bmatrix}
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| 1 & 2 \end{bmatrix}^{\mathrm{T}}
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| = \,
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| \begin{bmatrix}
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| 1 \\
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| 2 \end{bmatrix}
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| </math>
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| *<math>\begin{bmatrix}
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| 1 & 2 \\
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| 3 & 4 \end{bmatrix}^{\mathrm{T}}
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| =
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| \begin{bmatrix}
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| 1 & 3 \\
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| 2 & 4 \end{bmatrix}
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| </math>
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| * <math>
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| \begin{bmatrix}
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| 1 & 2 \\
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| 3 & 4 \\
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| 5 & 6 \end{bmatrix}^{\mathrm{T}}
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| =
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| \begin{bmatrix}
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| 1 & 3 & 5\\
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| 2 & 4 & 6 \end{bmatrix}
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| </math>
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| == Properties ==
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| For matrices '''A''', '''B''' and scalar ''c'' we have the following properties of transpose:
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| {{ordered list
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| |1= <math>( \mathbf{A}^\mathrm{T} ) ^\mathrm{T} = \mathbf{A} \quad \,</math>
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| :The operation of taking the transpose is an [[Involution (mathematics)|involution]] (self-[[Inverse matrix|inverse]]).
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| |2= <math>(\mathbf{A}+\mathbf{B}) ^\mathrm{T} = \mathbf{A}^\mathrm{T} + \mathbf{B}^\mathrm{T} \,</math>
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| :The transpose respects addition.
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| |3= <math>\left( \mathbf{A B} \right) ^\mathrm{T} = \mathbf{B}^\mathrm{T} \mathbf{A}^\mathrm{T} \,</math>
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| :Note that the order of the factors reverses. From this one can deduce that a [[square matrix]] '''A''' is [[Invertible matrix|invertible]] if and only if '''A'''<sup>T</sup> is invertible, and in this case we have ('''A'''<sup>−1</sup>)<sup>T</sup> = ('''A'''<sup>T</sup>)<sup>−1</sup>. By induction this result extends to the general case of multiple matrices, where we find that ('''A'''<sub>1</sub>'''A'''<sub>2</sub>...'''A'''<sub>''k''−1</sub>'''A'''<sub>''k''</sub>)<sup>T</sup> = '''A'''<sub>''k''</sub><sup>T</sup>'''A'''<sub>''k''−1</sub><sup>T</sup>...'''A'''<sub>2</sub><sup>T</sup>'''A'''<sub>1</sub><sup>T</sup>.
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| |4= <math>(c \mathbf{A})^\mathrm{T} = c \mathbf{A}^\mathrm{T} \,</math>
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| :The transpose of a [[Scalar (mathematics)|scalar]] is the same scalar. Together with (2), this states that the transpose is a [[linear map]] from the [[Vector space|space]] of {{nowrap|''m'' × ''n''}} matrices to the space of all {{nowrap|''n'' × ''m''}} matrices.
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| |5= <math>\det(\mathbf{A}^\mathrm{T}) = \det(\mathbf{A}) \,</math>
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| :The [[determinant]] of a square matrix is the same as that of its transpose.
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| |6= The [[dot product]] of two column [[vector space|vector]]s '''a''' and '''b''' can be computed as
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| :<math> \mathbf{a} \cdot \mathbf{b} = \mathbf{a}^{\mathrm{T}} \mathbf{b},</math>
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| which is written as '''a'''<sub>''i''</sub> '''b'''<sup>''i''</sup> in [[Einstein notation]].
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| |7= If '''A''' has only real entries, then '''A'''<sup>T</sup>'''A''' is a [[positive-semidefinite matrix]].
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| |8= <math>(\mathbf{A}^\mathrm{T})^{-1} = (\mathbf{A}^{-1})^\mathrm{T} \,</math>
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| : The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation '''A'''<sup>−T</sup> is sometimes used to represent either of these equivalent expressions.
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| |9= If '''A''' is a square matrix, then its [[Eigenvalue, eigenvector and eigenspace|eigenvalues]] are equal to the eigenvalues of its transpose since they share the same Characteristic polynomial.
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| }}
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| == Special transpose matrices ==
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| A square matrix whose transpose is equal to itself is called a [[symmetric matrix]]; that is, '''A''' is symmetric if
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| :<math>\mathbf{A}^{\mathrm{T}} = \mathbf{A} .</math>
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| A square matrix whose transpose is equal to its negative is called a [[skew-symmetric matrix]]; that is, '''A''' is skew-symmetric if
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| :<math>\mathbf{A}^{\mathrm{T}} = -\mathbf{A} .</math>
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| A square [[complex number|complex]] matrix whose transpose is equal to the matrix with every entry replaced by its [[complex conjugate]] is called a [[Hermitian matrix]] (equivalent to the matrix being equal to its [[conjugate transpose]]); that is, '''A''' is Hermitian if
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| :<math>\mathbf{A}^{\mathrm{T}} = \mathbf{A}^{*} .</math>
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| A square [[complex number|complex]] matrix whose transpose is equal to the negation of its complex conjugate is called a [[skew-Hermitian matrix]]; that is, '''A''' is skew-Hermitian if
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| :<math>\mathbf{A}^{\mathrm{T}} = -\mathbf{A}^{*} .</math> | |
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| A square matrix whose transpose is equal to its inverse is called an [[orthogonal matrix]]; that is, '''A''' is orthogonal if
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| :<math>\mathbf{A}^{\mathrm{T}} = \mathbf{A}^{-1} .</math>
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| == Transpose of a linear map ==
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| {{see also|Dual space#Transpose of a linear map|l1=Dual space'' (section ''Transpose of a linear map'')''}}
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| The transpose may be defined using a [[coordinate-free]] approach:
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| If {{nowrap|1=''f'' : ''V'' → ''W''}} is a [[linear operator|linear map]] between [[vector space]]s ''V'' and ''W'' with respective [[dual space]]s ''V''<sup>∗</sup> and ''W''<sup>∗</sup>, the ''transpose'' of ''f'' is the linear map {{nowrap|1=<sup>t</sup>''f'' : ''W''<sup>∗</sup> → ''V''<sup>∗</sup>}} that satisfies
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| :<math> {}^\mathrm{t} f (\phi ) = \phi \circ f \quad \forall \phi \in W^* .</math>
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| The definition of the transpose may be seen to be independent of any bilinear form on the vector spaces, unlike the adjoint ([[#Adjoint of a bilinear map|below]]).
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| If the matrix ''A'' describes a linear map with respect to [[basis (linear algebra)|bases]] of ''V'' and ''W'', then the matrix ''A''<sup>T</sup> describes the transpose of that linear map with respect to the dual bases.
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| === Transpose of a bilinear form ===
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| {{main|Bilinear form}}
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| Every linear map to the dual space {{nowrap|1=''f'' : ''V'' → ''V''<sup>∗</sup>}} defines a bilinear form {{nowrap|1=''B'' : ''V'' × ''V'' → ''F''}}, with the relation {{nowrap|1=''B''('''v''', '''w''') = ''f''('''v''')('''w''')}}. By defining the transpose of this bilinear form as the bilinear form <sup>t</sup>''B'' defined by the transpose {{nowrap|1=<sup>t</sup>''f'' : ''V''<sup>∗∗</sup> → ''V''<sup>∗</sup>}} i.e. {{nowrap|1=<sup>t</sup>''B''('''w''', '''v''') = <sup>t</sup>''f''('''w''')('''v''')}}, we find that {{nowrap|1=''B''('''v''','''w''') = <sup>t</sup>''B''('''w''','''v''')}}.
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| === Adjoint ===
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| {{distinguish|Hermitian adjoint}}
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| If the vector spaces ''V'' and ''W'' have respective [[nondegenerate form|nondegenerate]] [[bilinear form]]s ''B''<sub>''V''</sub> and ''B''<sub>''W''</sub>, a concept closely related to the transpose – the ''adjoint'' – may be defined:
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| If {{nowrap|1=''f'' : ''V'' → ''W''}} is a [[linear map]] between [[vector space]]s ''V'' and ''W'', we define ''g'' as the ''adjoint'' of ''f'' if {{nowrap|1=''g'' : ''W'' → ''V''}} satisfies
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| :<math>B_V(v, g(w)) = B_W(f(v),w) \quad \forall\ v \in V, w \in W .</math>
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| These bilinear forms define an [[isomorphism]] between ''V'' and ''V''<sup>∗</sup>, and between ''W'' and ''W''<sup>∗</sup>, resulting in an isomorphism between the transpose and adjoint of ''f''. The matrix of the adjoint of a map is the transposed matrix only if the [[basis (linear algebra)|bases]] are orthonormal with respect to their bilinear forms. In this context, many authors use the term transpose to refer to the adjoint as defined here.
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| The adjoint allows us to consider whether {{nowrap|1=''g'' : ''W'' → ''V''}} is equal to {{nowrap|1=''f''<sup> −1</sup> : ''W'' → ''V''}}. In particular, this allows the [[orthogonal group]] over a vector space ''V'' with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps {{nowrap|''V'' → ''V''}} for which the adjoint equals the inverse.
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| Over a complex vector space, one often works with [[sesquilinear form]]s (conjugate-linear in one argument) instead of bilinear forms. The [[Hermitian adjoint]] of a map between such spaces is defined similarly, and the matrix of the Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal.
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| ==Implementation of matrix transposition on computers==
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| On a [[computer]], one can often avoid explicitly transposing a matrix in [[Random access memory|memory]] by simply accessing the same data in a different order. For example, [[software libraries]] for [[linear algebra]], such as [[BLAS]], typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement.
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| However, there remain a number of circumstances in which it is necessary or desirable to physically reorder a matrix in memory to its transposed ordering. For example, with a matrix stored in [[row-major order]], the rows of the matrix are contiguous in memory and the columns are discontiguous. If repeated operations need to be performed on the columns, for example in a [[fast Fourier transform]] algorithm, transposing the matrix in memory (to make the columns contiguous) may improve performance by increasing [[memory locality]].
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| {{Main|In-place matrix transposition}}
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| Ideally, one might hope to transpose a matrix with minimal additional storage. This leads to the problem of transposing an ''n'' × ''m'' matrix [[in-place]], with [[Big O notation|O(1)]] additional storage or at most storage much less than ''mn''. For ''n'' ≠ ''m'', this involves a complicated [[permutation]] of the data elements that is non-trivial to implement in-place. Therefore efficient [[in-place matrix transposition]] has been the subject of numerous research publications in [[computer science]], starting in the late 1950s, and several algorithms have been developed.
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| ==See also==
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| *[[Invertible matrix]]
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| *[[Moore–Penrose pseudoinverse]]
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| *[[Projection (linear algebra)]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/detail/lecture05.htm MIT Linear Algebra Lecture on Matrix Transposes]
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| *[http://mathworld.wolfram.com/Transpose.html Transpose], mathworld.wolfram.com
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| *[http://planetmath.org/encyclopedia/Transpose.html Transpose], planetmath.org
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| *[http://khanexercises.appspot.com/video?v=2t0003_sxtU Khan Academy introduction to matrix transposes]
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| {{linear algebra}}
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| [[Category:Matrices]]
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| [[Category:Abstract algebra]]
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| [[Category:Linear algebra]]
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| [[de:Matrix (Mathematik)#Die transponierte Matrix]]
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It is time to address the slow computer issues whether or not we do not recognize how. Just because your computer is working thus slow or keeps freezing up; refuses to imply to not address the problem plus fix it. You may or could not be aware that any computer owner must know that there are certain points which your computer requires to maintain the greatest performance. The sad truth is the fact that numerous individuals whom own a system have no idea which it needs routine maintenance really like their vehicles.
Before actually getting the software it really is best to check on the companies that make the software. If you may discover details found on the type of standing each firm has, perhaps the risk of malicious programs will be reduced. Software from reputed companies have helped me, and various other consumers, to make my PC run faster.. If the product description does not look good to you, does not include details about the software, refuses to include the scan functions, we should go for another 1 which ensures you're paying for what we wish.
If you compare registry cleaners you require a fast acting registry cleaning. It's no superior spending hours and your PC waiting for a registry cleaning to complete its task. We wish the cleaner to complete its task inside minutes.
Paid registry cleaners on the other hand, I have found, are usually inexpensive. They supply usual, free updates or at least cheap changes. This follows considering the software manufacturer requires to confirm their product is most effective in staying ahead of its competitors.
Another common cause of PC slow down is a corrupt registry. The registry is a surprisingly important component of computers running on Windows platform. When this gets corrupted a PC will slowdown, or worse, not start at all. Fixing the registry is easy with all the use of a system and registry reviver.
The first thing we should do is to reinstall any program which shows the error. It's typical for many computers to have certain programs which need this DLL to show the error whenever you try plus load it up. If you see a particular program show the error, you must first uninstall which system, restart your PC and then resinstall the system again. This should substitute the damaged ac1st16.dll file plus remedy the error.
The disk requires room in purchase to run smoothly. By freeing up certain area from your disk, you are able to speed up the PC a bit. Delete all file in the temporary internet files folder, recycle bin, obvious shortcuts plus icons from the desktop which we do not utilize plus remove programs we do not employ.
Most folks make the mistake of trying to fix Windows registry by hand. I strongly recommend you don't do it. Unless you're a computer expert, I bet you'll invest hours plus hours learning the registry itself, let alone fixing it. And why if you waste a valuable time in learning plus fixing anything you understand nothing about? Why not let a smart and specialist registry cleaner do it for we? These software programs would be able to do the job inside a far better technique! Registry cleaners are quite affordable because well; you pay a 1 time fee and utilize it forever. Also, most professional registry cleaners are truly reliable plus effortless to use. If you require more info on how to fix Windows registry, just visit my website by clicking the link below!