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Unformatted text preview: Lecture 17 Today we begin studying the material that is also found in the Multilinear Algebra Notes. We begin with the theory of tensors . 4.3 Tensors Let V be a ndimensional vector space. We use the following notation. Notation. V k = V V . (4.14) k times For example, V 2 = V V, (4.15) V 3 = V V V. (4.16) Let T : V k R be a map. Definition 4.1. The map T is linear in its i th factor if for every sequence v j V, 1 j n, j = i , the function mapping v V to T ( v 1 , . . . , v i 1 , v, v i +1 , . . . , v k ) is linear in v . Definition 4.2. The map T is klinear (or is a ktensor ) if it is linear in all k factors. Let T 1 , T 2 be ktensors, and let 1 , 2 R . Then 1 T 1 + 2 T 2 is a ktensor (it is linear in all of its factors). So, the set of all ktensors is a vector space, denoted by L k ( V ), which we sometimes simply denote by L k . Consider the special case k = 1. The the set L 1 ( V ) is the set of all linear maps : V R . In other words, 1 L ( V ) = V . (4.17) We use the convention that L ( V ) = R . (4.18) Definition 4.3. Let T i L k i , i = 1 , 2, and define k = k 1 + k 2 . We define the tensor product of T 1 and T 2 to be the tensor T 1 T 2 : V k R defined by T 1 T 2 ( v 1 , . . . , v k ) = T 1 ( v 1 , . . . , v k 1 ) T 2 ( v k 1 +1 , . . . , v k ) . (4.19) We can conclude that T 1 T 2 L k . We can define more complicated tensor products. For example, let T i L k i , i = 1 , 2 , 3 , and define k = k 1 + k 2 + k 3 . Then we have the tensor product T 1 T 2 T 3 ( v 1 , . . . , v k ) = T 1 ( v i , . . . , v k 1 ) T 2 ( v k 1 +1 , . . . , v k 1 + k 2 ) T 3 ( v k 1 + k 2 +1 , . . . , v k ) . ....
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 Fall '04
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 Algebra, Vector Space

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