lecture17

# lecture17 - Lecture 17 Today we begin studying the material...

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Unformatted text preview: Lecture 17 Today we begin studying the material that is also found in the Multi-linear Algebra Notes. We begin with the theory of tensors . 4.3 Tensors Let V be a n-dimensional 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 k-linear (or is a k-tensor ) if it is linear in all k factors. Let T 1 , T 2 be k-tensors, and let λ 1 , λ 2 ∈ R . Then λ 1 T 1 + λ 2 T 2 is a k-tensor (it is linear in all of its factors). So, the set of all k-tensors 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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lecture17 - Lecture 17 Today we begin studying the material...

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