lecture17

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 1 , 2 R . Then 1 T 1 + 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 ) . ....
View Full Document

Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online