Let vj = (vj1,. . . . . . . . .,Vjn) Rn, j = 1,.

Let vj = (vj1,. . . . . . . . .,Vjn) ˆˆ Rn, j = 1,. . . . . . ., n, be fixed. The parallelepiped determined by the vectors vj is the set

and the determinant of the vj's is the number
det(v1...,vn) := del[vjk]n × n.
Prove that
Vol(P(v1...,vn)) = |det(v1,...,vn)|.
Check this formula for n = 2 and n = 3 to see that it agrees with the classical formulas for the area of a parallelogram and the volume of a parallelepiped.

Transcribed Image Text:

P(v1..... Vn) = (Vi ++ I, Vn : tj € [0, 1]),