Do drop a line to Caveman Colin for friendly hints/tips in answering these questions. If you send in some correct answers he might even send you a list of his favourite Stone-Age web sites!

• 1. Prove that if, for two vectors a and b, that if a is perpendicular to b then a - b has the same magnitude as a + b. What is the geometrical interpretation of this?

• 2. Prove that a+b is perpendicular to a-b if and only if |a| = |b|. What is the geometrical interpretation of this?

• 3. If a, b are the position vectors of A and B respectively, prove that the area of triangle OAB is
  Sqrt[(a.a)(b.b) - (a.b)²)]/2

• 4. Prove that (a+b)⁴ - (a-b)⁴ = 8 a.b(a.a+b.b)

• 5. Operation * is defined on two vectors as follows:
  
  a * b = (a + b)/(a.b)
  Prove that:
  • (i) (a * b).a + (a * b).b = (a + b).(a + b)/(a.b)
  • (ii) (a * a).a = 2
  • (iii) (a * j).i = 1/(a * i).j
    where i and j are unit vectors such that i.j = 0

• 6. The 'commutator' of two functions is defined as follows:
  [f,g](x) = fg(x) - gf(x)
  Prove that [[f,g],h](x) + [[h,f],g](x) + [[g,h],f](x) = 0

• 7. If f(x) = x/(x-1)
  • (i) Prove: fⁿ(x) = x if n is even
  • (ii) Prove: fⁿ(x) = f(x) if n is odd

• 8. Prove that any matrix representing an enlargement (center O) commutes with any representing a rotation, where the transformations are in two dimensions.

• 9. Prove that only a matrix of the form:
  

   a b
  -b a
  

  will commute with the general 2-D rotation matrix (rotations about O)

• 10. The commutator of two matrices M, N is defined as:
  [M,N] = MN - NM
  Defining

  M =  1 k    R = cos(A) -sin(A)    N = 1  0
       0 1        sin(A)  cos(A)        0 -1
  

  and I as the 2x2 identity matrix, prove that:
  • (i) [M,R] = ksin(A) N
  • (ii) [N,M] = 2M-I
  Give a geometrical meaning to M, R, N, [N,R]

• 11. Matrices are defined as follows (i² = -1)

  A =  1  0    B = 0  1    N = 0  -i
       0 -1        1  0        i   0
  

  and I as the 2x2 identity matrix, prove that:
  • (i) A² = B² = N² = I
  • (ii) The determinant of AB + BN + NA is 3