Index Notation and Dual Vectors

[BBeltGrl]

Homework Statement

Find the dual vector of the following tensor:

6 3 1
4 0 5
1 3 2

Homework Equations

dj= EijkTik

Where Eijk = 1 if ijk=123, 231, 312
Eijk = 0 if i=j i=k or j=k
Eijk = -1 if ijk = 132, 213, 321

The Attempt at a Solution

Ok, so I'm not really sure how to solve this but my thought is that it is simply multiply diagonally across the matrix such that

dj= (6*0*2)*(1) + (3*5*1)(1) +(1*4*3)*(1) + (1*0*1)(-1) + (6*5*3)(-1) +(3*4*2)(-1)

However, I don't see how that gets me a vector in the end.

Please HELP!

 

[Dick]

dj is the jth component of the dual vector. So there are three different values d1, d2 and d3. Just put j=1,2,3 and find them.