7—Operators and Matrices 2027.6 You have a mass attached to four springs in a plane and that are in turn attached to four walls, the mass is
at equilibrium. Two opposing spring have spring constantk 1 and the other two arek 2. Push on the mass with a
forceF~ and the resulting displacement ofmisd~=f(F~), defining a linear operator. Compute the components
offin an obvious basis and check a couple of special cases to see if the displacement is in a plausible direction,
especially if the twok’s are quite different.
7.7 On the vector space of quadratic polynomials, degree≤ 2 , the operatord/dxis defined: the derivative of
such a polynomial is a polynomial. (a) Use the basis~e 0 = 1,~e 1 =x, and~e 2 =x^2 and compute the components
of this operator. (b) Compute the components of the operatord^2 /dx^2. (c) Compute the square of the first
matrix and compare it to the result for (b).
7.8 Repeat the preceding problem, but look at the case of cubic polynomials, a four-dimensional space.
7.9 In the preceding problem the basis 1 ,x,x^2 ,x^3 is too obvious. Take another basis, the Legendre polynomials:
P 0 (x) = 1, P 1 (x) =x, P 2 (x) =3
2
x^2 −1
2
, P 3 (x) =5
2
x^3 −3
2
xand repeat the problem, finding components of the first and second derivative operators. Verify an example
explicitly to check that your matrix reproduces the effect of differentiation on a polynomial of your choice. Pick
one that will let you test your results.
7.10 What is the determinant of the inverse of an operator, explaining why?
7.11 Eight identical point massesmare places at the corners of a cube that has one corner at the origin of the
coordinates and has its sides along the axes. The side of the cube is length=a. In the basis that is placed along
the axes as usual, compute the components of the inertia tensor.
Ans:I 11 = 8ma^2
7.12 For the dumbbell rotating about the off-axis axis in Eq. ( 16 ), what is the time-derivative ofL~? In very
short timedt, what new direction doesL~take and what then isdL~? That will tell youd~L/dt. Prove that this is
~ω×L~.