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**Unformatted text preview: **MATH 601, AUTUMN 2008 Volume forms 1. Definitions. Given vector spaces V 1 ,...,V r ,W over a scalar field K , by a W-valued r-factor multiplication , or a W-valued r-linear function on V 1 ,...,V r we mean any function B assigning to arbitrary v 1 V 1 ,...,v r V r a value B ( v 1 ,...,v r ) W that depends linearly on each variable v 1 ,...,v r separately. When r = 1 , 2 , 3, one speaks of linear, bilinear and trilinear functions, and when r is not specified, the term multilinear is used. When all spaces V 1 ,...,V r are the same: V 1 = ... = V r = V , a multiplication B as above is called symmetric or, respectively, skew-symmetric if the value B ( v 1 ,...,v r ) remains unchanged (or, respectively, changes sign) whenever two neighboring entries in the r-tuple ( v 1 ,...,v r ) are switched. By a volume form in a real or complex vector space V of finite dimension n &gt; 0 we mean any scalar-valued skew-symmetric n-linear function on V that is not identically zero. In dimension 2 one uses the phrase area form . 2. Existence, uniqueness and properties. In a real or complex vector space V of any positive finite dimension, a volume form always exists, and is unique up to multiplication by a nonzero scalar. Furthermore, if dim V = n , and e 1 ,...,e n is any basis of V , a volume form can be uniquely defined by prescribing the scalar ( e 1 ,...,e n ) (which can have any nonzero value). Finally, given a volume form in a space V of dimension n &gt; 0, a system v , ...,v n of vectors in V is linearly dependent if and only if ( v...

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