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In linear algebra, the column space of a matrix A is the span of its column vectors. The column space of a matrix is the range of the corresponding matrix transformation. Let F be a field; the column space of an m × n matrix with components from F is a linear subspace of the m-space F m. The dimension of the column space is at most min. A definition for matrices over a ring K is possible; the row space is defined similarly. This article considers matrices of real numbers; the row and column spaces are subspaces of Rm respectively. Let A be an m-by-n matrix. Rank = dim = dim, rank = number of pivots in any echelon form of A, rank = the maximum number of linearly independent rows or columns of A. If one considers the matrix as a linear transformation from Rn to Rm the column space of the matrix equals the image of this linear transformation; the column space of a matrix A is the set of all linear combinations of the columns in A. If A = colsp = span; the concept of row space generalizes to matrices over C, the field of complex numbers, or over any field.

Intuitively, given a matrix A, the action of the matrix A on a vector x will return a linear combination of the columns of A weighted by the coordinates of x as coefficients. Another way to look at this is that it will first project x into the row space of A, perform an invertible transformation, place the resulting vector y in the column space of A, thus the result y = A x must reside in the column space of A. See singular value decomposition for more details on this second interpretation. Given a matrix J: J = the rows are r1 =, r2 =, r3 =, r4 =; the row space of J is the subspace of R5 spanned by. Since these four row vectors are linearly independent, the row space is 4-dimensional. Moreover, in this case it can be seen that they are all orthogonal to the vector n =, so it can be deduced that the row space consists of all vectors in R5 that are orthogonal to n. Let K be a field of scalars. Let A be an m × n matrix, with column vectors v1, v2... vn. A linear combination of these vectors is any vector of the form c 1 v 1 + c 2 v 2 + ⋯ + c n v n, where c1, c2... cn are scalars.

The set of all possible linear combinations of v1, ... ,vn is called the column space of A. That is, the column space of A is the span of the vectors v1, ... , vn. Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector: A = = = c 1 [ a 11 ⋮ a m

Jules Asner is a screenwriter and author as well as a former entertainment journalist, television personality and model. Asner was born Julie Ann White in Arizona. Aged 16, she was discovered at a modeling convention in Arizona, she began her career as an Elite model. She shared a bunkroom with Cindy Crawford during the early years of her modeling career. In 1986, she graduated from McClintock High School in Tempe, she attended college at the University of Los Angeles. Asner is known as a long-time E! personality. During her peak with the network she hosted six hours of programming daily. Among the series she hosted were Revealed with Jules Asner, E! News Live and Wild On! as well as many live specials for the network. She received a Prism Award as well as a Gracie Award for her interview show Revealed with Jules Asner. Prior to joining E! Asner was an entertainment reporter for Reuters Television Good Morning England, The Entertainment Show on Sky Television, Extra, she hosted Live by Request on A&E and was a segment director and producer on now defunct tabloid show, Hard Copy.

Miramax Books purchased the publishing rights to Whacked. The book was released June 3, 2008. Whacked was a top ten fiction bestseller on Amazon, was in its third printing within its first month of release. According to The Playlist, sources confirmed that Asner wrote the screenplay for Soderbergh's film Logan Lucky, she chose to use the pseudonym Rebecca Blunt because she did not want the'story' surrounding the film to be that "Soderbergh was directing his wife's script." In 2018 a Twitter account belonging to TheRebeccaBlunt posted photos of Asner and Soderbergh behind the scenes working on Logan Lucky. Asner did uncredited rewrite work on Magic Mike and several of his other films, she married film director Steven Soderbergh in their New York apartment on May 10, 2003. Soderbergh credits his wife with influencing his female characters. Jules Asner on IMDb Jules Asner 2003 Interview on Sidewalks Entertainment

Freda Payne Sings the I Hate Barney Songbook: A Parody is a comedy album by Freda Payne, released in 1995 and is one of three of her albums recorded for the label Dove. This album is one of the many contributions to Anti-Barney humor and some of the songs on this album are parodies of songs of the television show Barney & Friends, including "I Love You," "Barney Is a Dinosaur," "Good Manners," and "Please and Thank You." Produced By Tony Haynes Music recorded at the Groove Asylum Recording Studios, California Music engineer: Kenny Jones Vocals recorded at Interlok Studios, California Vocal engineer: Patrick Torres Produced by: Perry Botkin and Michael Viner Production manager: Karen Silverman Production coordinator: Cindy Jo Hinkleman Illustration by: Nate Butler Design: Bear Canyon Creative

Windows Phone Store is a former app store, developed by Microsoft for its Windows Phone. It was launched along with Windows Phone 7 in October 2010. With the rollout of Windows Phone 7.5, Microsoft unveiled the online Marketplace that offers silent, over the air installation of apps. In August 2012, Microsoft rebranded Windows Phone Marketplace to Windows Phone Store. In 2015, Microsoft announced that Windows Phone Store would be phased out and replaced by Windows Store which would act as a unified store for all Windows-powered devices; this process was complemented by the Apps on Windows website, an interim solution before the unified Windows Store. Windows Phone Store supported credit card purchases, operator billing, ad-supported content; the store featured a "try-before-you-buy" option, where the user had an option to download a trial or demo for a commercial app. Other features are said to be similar to Windows Phone Store's predecessor, Windows Marketplace for Mobile; the Windows Phone Store had 61 categories split up into 25 sub-categories.

Apps could only be placed in one category. Windows Phone Store featured downloads for 3D games that will have integrated Xbox Live connectivity and features; the ability to download a XAP file to a computer was present, allowing a user to install an app from their SD card if no internet access was available. Developers had to pay an annual subscription fee of \$99 to become an App Hub member and submit apps to the Windows Phone Store. This, according to Todd Brix, the General Manager for Windows Phone Apps and Store team, was on an ongoing promotion at \$19. There was a limit of 100 free submissions for free apps. Developers could choose to work with OEMs to deploy their apps; this process was used by Microsoft Mobile for apps specific to Lumia devices. OEM-exclusive apps were deployed and available to end-users in a separate channel available only on OEM-specific hardware. Most Windows Phone OEMs have a category, such as "Lumia Collection", "Samsung Zone", "HTC Apps" and "Huawei Selected" for this purpose.

A user could download apps from the Windows Phone Store. Microsoft had lined up a wide range of popular games to be available from the launch of Windows Phone 7. At Gamescom, Microsoft unveiled more than 50 premium Windows Phone 7 games and apps that used the Xbox Live mobile connection. Windows Phone Store grew swiftly since its launch and by February 2012, it had outgrown Blackberry App World with 70,000 apps available. In June 2012, after 20 months, Windows Phone Marketplace has reached 100,000 apps; the growth to achieve 100,000 apps was faster than Android with 24 months, but slower than iOS with 16 months. The number ramped up to 150,000 in December 2012, followed by 200,000 in December 2013. Windows Phone Store contained more than 300,000 apps in August 2014. Apps in Windows Phone Store are subjected to a content policy, which exists to guide app developers, to facilitate a restriction or banning of certain content. Examples of restricted or banned content include pornography, promotion of violence, hate, or the usage of drugs and tobacco.

Suggestions or depictions of prostitution, sexual fetishes, or anything that "a reasonable person would consider to be adult or borderline adult content" are forbidden from the marketplace after the unification with Windows Store. Windows Phone 7 development is based on Silverlight, XNA, the. NET Compact Framework; the primary tools used for development are Expression Blend. Excluding unlocked developer devices, Windows Phone 7 only runs mobile apps that have first been approved by Microsoft and are only available via Windows Phone Store. Developers can include an advertising model built into their apps. However, in some countries, the share is only 56.1% when the customer purchases with methods covered by the Commerce Expansion Adjustment. Students can submit apps free of charge through the DreamSpark program. Apps for Windows Phone 8 can be developed either in native code or managed code, using Visual C#, Visual Basic. NET and Visual C++. Microsoft has deprecated XNA in favor of DirectX for Windows Runtime.

The phone emulator in the SDK is changed to use Hyper-V, which requires at least Windows 8 Pro or Windows Server 2012, in addition to hardware-assisted virtualization. A subset of Windows Runtime objects allow code reuse on Windows 8 and later. A subset of the traditional Windows API is available. Use of third-party frameworks such as Unity is supported. For a time, developers could develop Windows Phone 7 apps using the Windows 7 SDK and they would still be compatible with Windows Phone 8. Windows Phone App Studio was a one stop source for rapid app building for Windows Phone. With couple of configuration steps, users can generate a production-ready app for Windows Store. Once all the configuration and look and feel is set, this online studio let you publish the app directly to Windows Phone Store. Windows Phone App Studio make it easy for non-developers to publish homebrew or small business apps without indulging them in source code or hiring a developer, it can be leveraged as a starting point for professional developers for boilerplate code generation.

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Colonel Sir George Ashley Maude, was a British Army officer and Crown Equerry to Queen Victoria. Colonel Sir George Ashley Maude was born in 1817, the son of Rev Hon. John Charles Maude of Enniskillen and Mary Cely Trevilian, daughter of William Cely Trevilian, his paternal grandparents were Cornwallis Maude, 1st Viscount Hawarden and his third wife, Anne Isabella Monck, sister of 1st Viscount Monck. Sir George was educated at the public school Sherborne and entered the Royal Horse Artillery in 1834, which he commanded as Colonel, he served in the battles of Alma and Balaklava, took part in the Siege of Sebastopol. As Lieutenant-Colonel he was Military Assistant to the Mission to the Coronation of Czar Alexander II of Russia, 7 September 1856, led by Lord Granville, he was Deputy-Inspector to the Royal Irish Constabulary between 1858 and 1859, appointed Crown Equerry of the Royal Mews in the Royal Household of the Sovereign of the United Kingdom in 1859. He was appointed Knight Commander of the Order of the Bath in 1887.

In 1845 Colonel Sir George Ashley Maude married Katherine Katinka, youngest daughter of Charles Beauclerk of St Leonard's Lodge, Sussex. Through her father, Katherine was the granddaughter of Lady Diana Spencer, daughter of Lt-General Charles Spencer, 3rd Duke of Marlborough. Sir George and Lady Maude had one daughter, he died in 1894

St John the Baptist's Church is in the village of Allington, England. It is a redundant Anglican parish church in the care of the Friends of Friendless Churches; the church is recorded in the National Heritage List for England as a designated Grade II listed building. The earliest parts of the church date from the 12th century, but only fragments of this remain, namely the chancel arch, a portion of a Norman door in the north wall. Most of the church was rebuilt between 1847 and 1851, the architect being the "priest-architect" Fr William Grey; the church was declared redundant on 1 February 2010, was vested in the Friends of Friendless Churches in the following year. The church is constructed in limestone with limestone dressings; the roof is tiled. Its plan consists of a nave and a chancel, a south porch that rises into a tower; the tower is in two stages with a pyramidal tiled roof. In the upper stage are two-light bell openings, at the summit is a crenellated parapet with gargoyles; the nave has doors on the north door containing the Norman fragments.

Along the sides of the nave are two- and three-light square-headed windows. In the chancel are two re-set lancet windows dating from the late 12th or early 13th century, a three-light east window. Inside the church, the wall of the nave is colour-washed; the interior of the chancel has stencil and freehand decoration executed in 1876 by Heaton and Bayne. This is in memory of rector of the church for over 60 years; the chancel is floored with encaustic tiles. It contains a piscina; the stained glass in the east window is by Heaton and Bayne, depicts the Crucifixion. The font dates from the 19th century, is in the style of the 12th century, it has a pyramidal pierced cover dating from the 17th century. The pulpit is in oak, dates from the 19th century. There are two pews dating from the 16th-17th century, pews designed to match them. In the porch is an oak churchwardens' coffer from the 16th-17th century with three locks, a stool dating from the 17th century. In the churchyard are two structures that have been listed at Grade II.

One of these is a pair of limestone coffin tombs dating from the 18th century. One of these is inscribed with the date 1728; the other consists of two chest tombs dating from the early 19th century. A forerunner of Revd Fulwar William Fowle was Revd Thomas Fowle, vicar between 1793 and 1797, engaged to Cassandra Austen, sister of Jane, but who died before he could marry her. Fowle died of yellow fever in the West Indies trying to raise sufficient funds for their marriage