 
Summary: How elementary linear maps change areas.
Fix an integer n 2. Let
C = {x Rn
: 0 xi 1, i = 1, . . . , n}.
The "C" here stands for cube. Our goal in this Introduction is to prove that
(1) L[C] = det L, L GL(Rn
).
For each c R {0} let Sc GL(Rn
) be defined by
Sc(x) = (x1, x2, . . . , xn + cxn1), x Rn
.
The 'S' here stands for shear; that this is reasonable terminology can be seen by drawing a picture of what
Sc does to the cube C. For each c > 0 let Dc GL(Rn
) be defined by
Dc(x) = (x1, x2, . . . , cxn), x Rn
.
The 'D' here stands for dilate. For each Rn
let P L(Rn
, Rn
) be defined by
