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Unformatted text preview: n n n n R R T : n R A T = x x x , ) ( . Suppose that det( A ) , that is T is an invertible transformation. From the definition of matrix multiplication, the j th column vector of AB is so j A b . The linear transformation T thus carries the original nbox [ n A A A AB b b b . . . . . . 2 1 = ] determined by the column vectors of B into a new nbox determined by the column vectors of AB . The new nbox has volume ) det( ) det( ) det( B A AB = . That is, the volume of the new nbox, or image box, is equal to V A ) det( . Thus ) det( A is referred to as the volumechange factor for the linear transformation T . This is a basic property that explains the change of variables in multiple integral in case of linear maps. Above as well as in the next part the standard notation T is used for a transformation. In your text the notation is G ....
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 Fall '09
 RomauldStanczak
 Calculus

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