Math 603-601 - Fall 2002

Assignment 2

1. In each of the following cases, find the matrix A for which
   [v]_D=A[v]_B
   In addition, find the D coordinates for the given vector v.
   1. B={1, 2x, 4x²-1} and D={1, x, x²}, where v is the polynomial p(x)=x(3-2x).
   2. B={(1,0,0)^T, (0,1,0)^T, (0,0,1)^T} and D={(1,0,-1)^T, (1,1,1)^T, (-1,2,1)^T}, where v is the column vector (in B coordinates) (1,-2,1)^T.

2. Let B = {v₁ ... v_n}, D = {w₁ ... w_n}, and E = {u₁ ... u_n} be bases for a vector space V. Suppose that you are given these column matrices:
   A₁ = [ [v₁]_E ... [v_n]_E]
   A₂ = [ [w₁]_E ... [w_n]_E]
   The columns of A₁ and A₂ are the E-coordinates of the bases B and D, respectively. Show that
   A₁[v]_B = A₂[v]_D,
   where v is an arbitrary vector in V.
3. Use the result from the previous problem to find the matrix that takes coordinates relative to B into ones relative to D if B = {1,2x-1,x²+x} and D = {1-x,x², x+1}. Hint: take E={1, x, x²}.

4. Let U be all functions f in C[0,1] that are linear between the points x = 0,1/10, 2/10,..., 1. (These are linear splines or "connect the dots" functions). Also, let T(x) = max(1 - 10|x|,0) be the tent function.
   1. Show that B = { T(x), T(x-1/10), T(x-2/10),..., T(x-1) } is a basis for U and that if f is in U then
      f(x) = f(0)T(x) + f(1/10)T(x-1/10) + f(2/10)T(x-2/10) +...+ f(1)T(x-1)
   2. Suppose that f is in U and that f(k/10) = sin(3k/10). Plot f and also plot sin(3x) on the interval x=0 to x=1. (Use your favorite software to do this.)