PROBLEM SET IV
DUE FRIDAY,  MARCH
For the next three problems, consider R4 (with standard basis (e0 , e1 , e2 , e3 )),
equipped with the bilinear form η given by the matrix


−1 0 0 0
 0 1 0 0

H=
 0 0 1 0 ,
0 0 0 1
so that η(v, w) = vT Hw. Consider the set L of linear transformations
M : R4 .
R4
such that for any v, w ∈ R4 ,
η(Mv, Mw) = η(v, w).
is is called the Lorentz group. Its elements are called Lorentz transformations.
Exercise . Show that all Lorentz transformations are invertible.
Exercise . Check that for any 3×3 orthogonal matrix R (i.e., a matrix R such
that RT R = id), the block matrix
(
)
1 0
,
0 R
is a Lorentz transformation.
Exercise . Write ϕ := tanh−1 (u). Prove that the matrix


cosh(ϕ) − sinh(ϕ) 0 0
− sinh(ϕ) cosh(ϕ) 0 0

Λϕ = 

0
0
1 0
0
0
0 1
is a Lorentz transformation. (is is the Lorentz boost in the positive x-direction
at speed u.)


DUE FRIDAY,  MARCH
Recall that the vector space Lin(Rm , Rn ) of linear maps Rm . Rn is isomorphic to the vector space Matn×m (R) ∼
= Rmn of n × m matrices. So we may
speak of open sets of Lin(Rm , Rn ).
Exercise . Show that the map
K : Lin(Rm , Rn ) × Lin(Rk , Rm ) .
given by composition (equivalently, the map
Rmn+km ∼
= Matn×m (R) × Matm×k (R) .
Lin(Rk , Rn )
Matn×k (R) ∼
= Rkn
given by matrix multiplication) is continuous.
Exercise . Consider the subset GL(Rn ) ⊂ Lin(Rn , Rn ) of the isomorphisms
Rn . Rn . (Equivalently, you may consider the subset GLn (R) ⊂ Matn×n (R)
of invertible n × n matrices.) Show that GL(Rn ) is an open set.
Exercise . List all the vector subspaces of Rm that are open subsets. List all
the vector subspaces of Rm that are closed subsets. Justify your answers.
Exercise . Consider the map g : R2 .
{ 2
g(x, y) :=
R given by the formula
x y
x4 +y2
if (x, y) ̸= (0, 0);
0
if (x, y) = (0, 0).
For any unit vector v ∈ R2 , compute limε→0 g(εv). Show that g is not continuous at the origin. [In the rst semester, we learned that a function R . R
was continuous at a point a ∈ R just in case the left-hand limit and the righthand limit both exist and are equal to the value of the function. Do things look
different here?]
Exercise . Consider the unit m-sphere in Rm+1 :
{
}
m
∑
Sm = (x0 , x1 , . . . , xm ) ∈ Rm+1 x2i = 1 .
i=0
m
Write down a homeomorphism R .
Recall that we may de
S − {(1, 0, . . . , 0)}.
m
ne an isomorphism M : Lin(Rm , Rn ) . Matn×m (R) by M(ϕ) :=
(ϕ(e1 ), ϕ(e2 ), . . . , ϕ(em )), where we think of the vectors ϕ(ei ) arranged vertically.