PROBLEM SET VII
DUE FRIDAY,  APRIL
Exercise . Consider the unit cube
[0, 1]m := {(x1 , x2 , . . . , xm ) ∈ Rm | ∀i ∈ {1, 2, . . . , m}, xi ∈ [0, 1]} ⊂ Rm .
Suppose φ : Rm .
Rm a linear map. What is the volume of the image of [0, 1]m under φ?
Exercise . Suppose m ≥ 2 an integer. For any distinct positive integers r, s ≤ n, and for any nonzero real number
α, let Ers (α) = (ers (α)ij ) denote the m × m matrix whose (i, j)-th entry is α if (i, j) = (r, s), and is the Kronecker
symbol otherwise:
{
α if (i, j) = (r, s)
ers (α)ij =
δ ij otherwise.
ese matrices are called elementary matrices. Show that multiplying a matrix M by an elementary matrix Ers (α) on
the right is the same as taking column r of M, multiplying it by α, and adding it to column s. Show that for any matrix
M, one may multiply M by elementary matrices on both the le and right to obtain a diagonal matrix. Deduce that
the determinant det : Matm×m . R is the unique map such that the following conditions hold.
() For any diagonal matrix diag[λ1 , λ2 , . . . , λm ], one has det diag[λ1 , λ2 , . . . , λm ] = λ1 λ2 · · · λm .
() For any elementary matrix Ers (α), one has det Ers (α) = 1.
() For any M, N ∈ Matm×m , one has det(MN) = det(M) det(N).
Exercise . e trace tr M of a square matrix M is the sum of the entries along the main diagonal. Show that for any
square matrix M, one has exp(tr(M)) = det(exp(M)).
Exercise . Compute the improper integral
∫ +∞
∫
2
I=
e−t dt = lim
+R
R→∞ −R
−∞
in the following manner. First, compute the integral
∫
e−(x
2 +y2 )
e−t dt
2
dx dy,
B(0,R)
using some convenient change of variables. Next, prove that
[∫
∫
2
2
e−(x +y ) dx dy =
−∞
[−R,R]×[−R,R]
Finally, check that
∫
lim
R→∞ [−R,R]×[−R,R]
e−(x
+∞
2 +y2 )
e−t dt
∫
dx dy = lim
R→∞ B(0,R)
and compute I.

2
]2
.
e−(x
2 +y2 )
dx dy,