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Linear independence

In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others. These concepts are central to the definition of dimension. A vector space can be of finite-dimension or infinite-dimension depending on the number of linearly independent basis vectors; the definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining a basis for a vector space. A sequence of vectors from a vector space V is said to be linearly dependent, if there exist scalars a 1, a 2, …, a k, not all zero, such that a 1 v 1 → + a 2 v 2 → + ⋯ + a k v k → = 0 →, where 0 → denotes the zero vector. Notice that if not all of the scalars are zero at least one is non-zero, say a 1, in which case this equation can be written in the form v 1 → = − a 2 a 1 v 2 → + ⋯ + − a k a 1 v k →. Thus, v 1 → is shown to be a linear combination of the remaining vectors.

A sequence of vectors is said to be linearly independent if the equation a 1 v 1 → + a 2 v 2 → + ⋯ + a n v n → = 0 →, can only be satisfied by a i = 0 for i = 1, …, n. This implies that no vector in the sequence can be represented as a linear combination of the remaining vectors in the sequence. In other words, a sequence of vectors is linearly independent if the only representation of 0 → as a linear combination of its vectors is the trivial representation in which all the scalars a i are zero. More concisely, a sequence of vectors is linear independent if and only if 0 → can be represented as a linear combination of its vectors in a unique way; the alternate definition, that a sequence of vectors is linearly dependent if and only if some vector in that sequence can be written as a linear combination of the other vectors, is only useful when the sequence contains two or more vectors. When the sequence contains no vectors or only one vector, the original definition is used. In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows.

More let V be a vector space over a field K, let be a family of elements of V. The family is linearly dependent over K if there exists a finite family of elements of K, all non-zero, such that ∑ j ∈ J a j v j = 0. A set X of elements of V is linearly independent if the corresponding family x∈X is linearly independent. Equivalently, a family is dependent if a member is in the closure of the linear span of the rest of the family, i.e. a member is a linear combination of the rest of the family. The trivial case of the empty family must be regarded as linearly independent for theorems to apply. A set of vectors, linearly independent and spans some vector space, forms a basis for that vector space. For example, the vector space of all polynomials in x over the reals has the subset as a basis. A geographic example may help to clarify the concept of linear independence. A person describing the location of a certain pla

Jean-Michel Byron

Jean-Michel Byron is a South African-born funk and rock vocalist. The singer-songwriter is best known for serving for a time as the lead vocalist for the rock band Toto for new material on their first greatest hits album, Past to Present, as well as the band's Greatest Hits Live...and More project. Byron was the fourth frontman in the band's history, replacing Joseph Williams after Williams had suffered from voice issues due to intense touring, the South African singer was pushed by the group's record label; the line-up change divided Toto's band-members, yet key group member Jeff Porcaro expressed support at first. His tenure with the band ended up being both brief and controversial with fans. In more recent years, Byron has collaborated with the jazz fusion group Michael Sanders & the One Tribe Nation. Byron's musical career began in 1983 as a session singer, having played in groups doing cover songs. From East London, South Africa, he attended high school at John Bissicker High School in a former "coloured area".

He sang at high school concerts and community shows. Byron joined Toto for their Past to Present 1977–1990 album. Singing on the release's four new tracks, including the popular funk-fueled song "Love Has the Power", Byron had a style mixing rock music with R&B different than the group's past vocalists, being directly influenced by artists such as George Michael; the album proved to be a hit after it came out in mid-1990, reaching top 20 spots on album ranking charts in over four different nations worldwide including Germany and Switzerland, it attracted critical praise. Reviewers had mixed remarks on Byron's vocal abilities, with AllMusic's William Ruhlmann finding the artist "more soulful than his predecessors, but no more memorable."The band's record label had pushed Byron on the other band members, who felt uncertain given the difference of styles between Byron's soul music background and Toto's previous rock work. Despite album success, things would soon become more intense. During the album's supporting tour in 1990, Byron and the band began to clash.

His on-stage antics and diva-like behavior irked other band members and had a polarizing influence on Toto's fans, some concertgoers flipping him off and heckling him during performances. "Byron starts dancing around. I’m looking at Jeff with bulging eyes. In rehearsal, Byron was just sitting there, but now he's out doing this Michael Jackson on crack shit, with a golf glove on one hand, my jaw was on the floor," Toto guitarist, co-vocalist Steve Lukather remarked, "We’re mortified... but it had all gone to his head". The group ended up bringing in additional singers, Steve Lukather took in a more prominent role until Byron ended up being relegated to the role of a backing vocalist. Upon Byron's dismissal, Toto were once again at a crossroads, having had an acrimonious divide during his tenure. Fans' anti-Byron sentiment had built up to the point that all of his songs were cut from the filmed Toto Live video from the aforementioned tour, though his studio work has remained a hot topic among critics and listeners alike over the years.

After being kicked out of Toto, Byron released a solo album titled Byron, which contains an alternate version of "Love Has the Power". More Byron has collaborated with the group "Michael Sanders & the One Tribe Nation", a band playing a mix of jazz rock with funk, world music, more, he appeared on the outfit's 2005 debut album Servants of a Lesser God. With Byron's material alongside works by Luis Conte and Andy Vargas of Santana among others, the release has received positive remarks from several critics such as Morley Seaver of antimusic.com, who praised the "hypnotic blend of musical virtuosity and melodic ear-pleasers" on the album. Byron contributed to the Star Jasmine Music Foundation project's 2007 release, titled Star Jasmine, with his material alongside songs performed by artists such as Ginger Baker and Van Morrison, his song was called "For a Dancer". The charitable organization assists Californians in many ways such as operating "JAMS", a volunteer-run music school giving instruments and lessons to at-risk youths in the Santa Barbara area.

1990 - Past to Present / Toto 2005 - Servants of a Lesser God / Michael Sanders & the One Tribe Nation

2015 Horizon League Men's Basketball Tournament

The 2015 Horizon League Men's Basketball Tournament began on March 3, 2015, with the championship game on Tuesday, March 10. First round games were played on the home court of the higher-seeded team; the second round and semifinals were hosted by Valparaiso, because the Crusaders won the regular season conference championship. Since Valparaiso won its semifinal game, it hosted the championship game. Only eight of the nine teams in the conference played in the Horizon League Tournament, as Milwaukee has been deemed ineligible for postseason play for the 2014–15 season. Teams were seeded by 2014–15 Horizon League season record, with a tiebreaker system to seed teams with identical conference records; the top two teams received a bye to the semifinals and the third and fourth seeds received byes to the quarterfinals. The tiebreaking system is. Record against the team occupying the highest position in the standings. Continue down through the standings until the tie is broken. Higher RPI based on Collegiate Basketball News.

Three or more teamsCombined record of each of the tied teams against the other teams involved in the tie until the tie is broken. Record against the team occupying the highest position in the standings. Continue down through the standings until the tie is reduced to a two-team tie. Use the two team system. Higher RPI based on Collegiate Basketball News. First round games at campus sites of higher seeds. Second round and semifinals hosted by No. 1 seed. Championship game hosted by highest remaining seed

Jacob M. Nachtigall

Jacob M. Nachtigall was an American architect active in Omaha and eastern Nebraska; the surname is sometimes spelled Nachtigal. He designed other buildings in Nebraska. Nachtigall was born in Germany in about 1874 and came with his family to the U. S. in 1883. He apprenticed under Thomas Rogers Kimball during 1900–1908 and went independent in 1909. A number of his works survive and are listed on the U. S. National Register of Historic Places. Works include: Father Flanagan House, off U. S. 6, Boys Town, NE, NRHP-listed Immaculate Conception Church and School, 1024 S. 24th St. Omaha, NE, NRHP-listed St. Anthony's Church and School, 514 W. Main St. and 103 N 6th St. Cedar Rapids, NE, NRHP-listed St. Bonaventure Church Complex, off NE 14, Raeville, NE, NRHP-listed St. Leonard's Catholic Church, 502–504 S. Nebraska St. Madison, NE, NRHP-listed St. Mary of the Assumption Catholic Church and Grottoes, 336 W. Pine St. Dwight, NE, NRHP-listed; the church, built in 1914, was designed by Nachtigal. St. Michael's Catholic Church NE of Greeley Ctr.

Spalding, NE, NRHP-listed Swartz Printing Company Building, 714 S. 15th St. Omaha, NE, NRHP-listed West Point City Auditorium, 237 N. Main St. West Point, NE, NRHP-listed

Jim Stanton

Jim Stanton is an American composer and political writer. Stanton was a drummer with the Tommy Dorsey Orchestra in the mid-1960s, he played with big bands through the 1960s and early 1970s. He co-led performing group, Dialogue, in the 1970s. Stanton wrote the score for an original radio production of The Trial of the Catonsville Nine by Daniel Berrigan, played on WBAI-FM in New York City in 1971, he wrote a two-person play, Chatoyant, in 1977, opening and performing it in the Philadelphia area with music played by Terry Gross on Fresh Air. Stanton composed sound and score for an original production of Shakespeare's The Tempest in 1980, he would go on to write many independent orchestral pieces, scored for many different instrumental combinations. In the spring of 2008, he began writing his autobiography. Stanton wrote many articles for the Philadelphia Weekly and various national publications on politics and history. Historycooperative.org Americanforeignrelations.com Proconservative.net

Progress M-18

Progress M-18 was a Russian cargo unmanned spacecraft, launched in 1993 to resupply the Mir space station. The thirty-sixth of sixty four Progress spacecraft to visit Mir, it used the Progress-M 11F615A55 configuration, had the serial number 218, it carried supplies including food and oxygen for the EO-13 crew aboard Mir, as well as equipment for conducting scientific research, fuel for adjusting the station's orbit and performing manoeuvres. Progress M-18 was launched at 06:41:47 GMT on 22 May 1993, atop a Soyuz-U2 carrier rocket flying from Site 1/5 at the Baikonur Cosmodrome, it was the last Progress spacecraft to be launched on a Soyuz-U2. Following two days of free flight, it docked with the Forward port of Mir's core module at 08:24:44 GMT on 24 May. During the 40 days for which Progress M-18 was docked, Mir was in an orbit of around 388 by 390 kilometres, inclined at 51.6 degrees. Progress M-18 undocked from Mir at 15:58:16 GMT on 3 July, it was deorbited around a day to a destructive reentry over the Pacific Ocean.

Before undocking, a VBK-Raduga capsule launched aboard Progress M-17 had been installed on Progress M-18, this separated once the deorbit burn was complete. The capsule landed at 17:13 GMT. 1993 in spaceflight List of Progress flights List of unmanned spaceflights to Mir