Exercise [13.40]

Yuanyuan · 23 Jul 2010, 23:43

Part 1

Suppose $\{e^i\}$ is a basis for V^*

, then $\{e^i \tensor e^j \}$ is a basis for $V^* \tensor V^*$ . All symmetric tensors in $V^* \tensor V^*$ could be written as $\a_{ij}e^i \tensor e^j$ , where $a_{ij} = a{ji}$ , there are totally n(n-1)/2 +n = n(n+1)/2

such symmetric tensors.

A symmetric tensor could be naturally identify with a symmetric matrix

[a^1_1, a^1_2, a^1_3 ...]<br />[a^1_2, a^2_2, a^2_3 ...]<br />[a^1_3, a^2_3, a^3_3 ...]