Find 4-D Volume of R^7 Vector in R^7

[calvino]

I have to find the 4-dimensional volume of {s'v' + s''v'' + s'''v''' +s''''v'''': 0<= s',s'',s''',s'''' <=1 }

in R^7, where v' = (1,1,0,1,0,0,1)^T,
v''=(1,0,0,1,1,0,0)^T
v'''=(0,0,1,1,1,0,0)^T
v''''=(0,0,0,0,1,1,1)^T


So I decided to try and take the determinant of the matrix which holds these vectors, by expanding along the 1st row consisting of elementary coordinates.

| e1 e2 e3 e4 e5 e6 e7 |
| 1 1 0 1 0 0 1 |
| 1 0 0 1 1 0 0 |
| 0 0 1 1 1 0 0 |
| 0 0 0 0 1 1 1 |


But, ummm... as you might already know, it doesn't work out. What do I do?

 

[calvino]

clarifying, all the s,v are scalars, vectors respecively...by "^T", i just meant to write transpose...