lecture19

lecture19 - Lecture 19 We begin with a review of tensors...

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 19 We begin with a review of tensors and alternating tensors. We defined L k ( V ) to be the set of k-linear maps T : V k R . We defined → 1 , . . . , e ∗ e 1 , . . . , e n to be a basis of V and e ∗ n to be a basis of V ∗ . We also defined k e ∗ I = e ∗ i 1 ⊗ ··· ⊗ e ∗ i k } to be a basis of L ( V ), where I = ( i 1 , . . . , i k ) , 1 ≤ i r ≤ n is a { k multi-index. This showed that dim L k = n . and T ∈ L k We defined the permutation operation on a tensor. For σ ∈ S n , we defined T σ ∈ L k by T σ ( v 1 , . . . , v k ) = T ( v σ − 1 (1) , . . . , v σ − 1 ( k ) ). Then we defined that T is alternating if T σ = ( − 1) σ T . We defined A k = A k ( V ) to be the space of all alternating k-tensors. We defined the alternating operator Alt : L k → A k by Alt ( T ) = ( − 1) σ T σ , and we defined ψ I = Alt ( e ∗ I ), where I = ( i 1 , . . . , i k ) is a strictly increasing multi-index. We proved the following theorem: Theorem 4.24. The ψ I ’s (where I is strictly increasing) are a basis for A k ( V ) . Corollary 6. If 0 ≤ k ≤ n , then n n ! dim A k = = k !( n − k )! . (4.69) k Corollary 7. If k > n , then A k = . { } We now ask what is the kernel of Alt ? That is, for which T ∈ L k is Alt ( T ) = 0?...
View Full Document

This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.

Page1 / 4

lecture19 - Lecture 19 We begin with a review of tensors...

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