MAT 203/222, Section 01
Assignment 03
Total points: 20 (5 each)
Due date: 21 April 11:59 PM
Question 1:
Let V = {Set of all invertible N × N matrices} with the standard definition of matrix addition
and scalar multiplication. Is V a vector space? If not, which axiom/axioms fails/fail?
Question 2:
Three frequently used coordinates in R3 are 1: Cartesian Coordinates, 2: Cylindrical coordinates, and 3: Spherical Coordinates. Unit vectors in these coordinates are the following.
Cartesian coordinates (x, y, z):
êx = î,
(1)
êy = ĵ,
(2)
êz = k̂.
(3)
Cylindrical coordinates (ρ, φ, z):
êρ = cos φ î + sin φ ĵ,
(4)
êφ = − sin φ î + cos φ ĵ,
(5)
êz = k̂
(6)
Spherical coordinates (r, θ, φ):
êr = sin θ cos φ î + sin θ sin φ ĵ + cos θ k̂,
(7)
êθ = cos θ cos φ î + cos θ sin φ ĵ − sin θ k̂,
(8)
êφ = − sin φ î + cos φ ĵ.
(9)
a) Find the expressions of (êr , êθ , êφ ) in terms of (êρ , êφ , êz ).
b) A vector can be expressed in terms of any basis. In particular, we can write:
V~ = Vx êx + Vy êy + Vz êz = Vρ êρ + Vφ êφ + Vz êz = Vr êr + Vθ êθ + Vφ êφ .
In terms of column vector notation we write
Vx
Vρ
Vr
VCr = Vy , VCl = Vφ , VSp = Vθ .
Vz
Vz
Vφ
(10)
(11)
The relations between them can be expressed in terms of matrix:
VCr = PCr←Cl VCl , VCr = PCr←Sp VSp ,
VCl = PCl←Cr VCr , VCl = PCl←Sp VSp ,
VSp = PSp←Cr VCr , VSp = PSp←Cl VCl ,
(12)
(13)
(14)
where PCr←Cl , PCr←Sp , PCl←Cr , PCl←Sp , PSp←Cr , PSp←Cl are all 3×3 matrices. Without explicitly
finding the matrices show that:
PSp←Cr PCr←Sp = I3 , PSp←Cl = PSp←Cr PCr←Cl .
c) Now explicitly compute PSp←Cl and show that PSp←Cl ∈ SO(3).
1
(15)
