- #1
coquelicot
- 299
- 67
Hello,
In R^3, the surface of the parallelogram determined by two vectors u and v is given by the norm of the cross product of u and v. For my research, I have to know if this can be generalized in the following manner:
Let e_1,..,e_n be the canonical basis of R^n, and Ext_k be the exterior algebral of rank k over R^n (k<=n). We assume that the basis e_1^..e^k, e_1^..^e_{k-1}^e_{k+1}, ..., e_2^..e_{k+1},... of Ext_k is oriented positively, and we endow it with the canonical scalar product. Now, is the following assertion true: the k-dimensional volume of the polytope determined by k vectors u_1,...,u_k is equal to the norm of u_1^...^u_k, computed with respect to the scalar product in the above basis.
When trying to prove this theorem, I have come to some other theorem that implies it easily (any suggestion will be appreciated). To begin with, consider again the case of R^3, and let A be an orthonormal matrix of R^3. Then it should be clear for a physisist that for two vectors u and v of R^3, Au^Av = det(A) A (u^v) (here ^ is the usual cross product).
Now, with the above notations for the bases, scalar product etc., is the following true:
if u_1,...,u_k are k vectors of R^n and if A is an orthonormal matrix of R^n, then Au_1^...^Au_k = (-1)^n det(A) B (u_1^...^u_k), where B is a rotation matrix of Ext_k with respect to the basis above. Furthermore, if k=n-1, B=Refl(A), where the columns of Refl(M) are equal to the columns of M in the reverse order.
thx.
In R^3, the surface of the parallelogram determined by two vectors u and v is given by the norm of the cross product of u and v. For my research, I have to know if this can be generalized in the following manner:
Let e_1,..,e_n be the canonical basis of R^n, and Ext_k be the exterior algebral of rank k over R^n (k<=n). We assume that the basis e_1^..e^k, e_1^..^e_{k-1}^e_{k+1}, ..., e_2^..e_{k+1},... of Ext_k is oriented positively, and we endow it with the canonical scalar product. Now, is the following assertion true: the k-dimensional volume of the polytope determined by k vectors u_1,...,u_k is equal to the norm of u_1^...^u_k, computed with respect to the scalar product in the above basis.
When trying to prove this theorem, I have come to some other theorem that implies it easily (any suggestion will be appreciated). To begin with, consider again the case of R^3, and let A be an orthonormal matrix of R^3. Then it should be clear for a physisist that for two vectors u and v of R^3, Au^Av = det(A) A (u^v) (here ^ is the usual cross product).
Now, with the above notations for the bases, scalar product etc., is the following true:
if u_1,...,u_k are k vectors of R^n and if A is an orthonormal matrix of R^n, then Au_1^...^Au_k = (-1)^n det(A) B (u_1^...^u_k), where B is a rotation matrix of Ext_k with respect to the basis above. Furthermore, if k=n-1, B=Refl(A), where the columns of Refl(M) are equal to the columns of M in the reverse order.
thx.