A set of at most d + 1 points in general linear position is also said to be " affinely independent " ( this is the affine analog of linear independence of vectors, or more precisely of maximal rank ), and d + 1 points in general linear position in affine " d "-space are an affine basis.
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:: : : And if you want to see the linear independence of the above v _ \ alpha algebrically, the fact is that n of them are already independent in the projection over the first n coordinates k = 0, 1, . ., n-1 : use the Vandermonde matrix ( and note that the wiki-article has it already written with the \ alpha _ i . ) .-- talk ) 23 : 26, 25 September 2009 ( UTC)
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When the vector space is finite-dimensional, for instance I = \ { 0, 1, \ ldots n-1 \ } with n > 0, the functions \ ell in the set " L " of the " outer condition " exactly are the ones that provide the spanning and linear independence properties with linear combinations \ ell ( b ) = c _ 0 b _ 0 + c _ 1b _ 1 + \ ldots c _ { n-1 } b _ { n-1 } and present generator property becomes the spanning one.
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To see this as an instance of the basis conjecture, one may use either linear independence of the vectors ( " x i ", " y i ", 1 ) in a three-dimensional real vector space ( where ( " x i ", " y i " ) are the Cartesian coordinates of the triangle vertices ) or equivalently one may use a matroid of rank three in which a set " S " of points is independent if either | " S " | d " 2 or " S " forms the three vertices of a non-degenerate triangle.
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Beginning with a collection of'units "'e 1, e 2, e 3, . . . ", he effectively defines the free linear space which they generate; that is to say, he considers formal linear combinations " a 1 e 1 + a 2 e 2 + a 3 e 3 + . . . " where the " a j " are real numbers, defines addition and multiplication by real numbers [ in what is now the usual way ] and formally proves the linear space properties for these operations . . . . He then develops the theory of linear independence in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts.
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