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)