Math 603-601 - Fall 2002

Assignment 1

1. Verify that the set U of all twice continuously differentiable solutions on [a,b] to the ODE y'' + p(x)y' + q(x)y = 0, where p and q are arbitrary continuous functions on [a,b], is a subspace of C²[a,b]. (Don't try to solve the equation, because you can't! The problem is an application of our theorem on subspaces.)

2. Let P_n be polynomials of degree n or less in x.
   1. Show that P₂ is a subspace of P₃
   2. Let U be the subset of P_n comprising all polynomials such that p(1)+p'(1)=0 and p(2)=0. Is U a subspace? What happens if the condition is changed to p(1)+p'(1)=0 or p(2)=0?

3. Determine whether the following sets of vectors are LI or LD.
   1. { 1-x, 1+2x,1+x}, V = P₁
   2. { cos(2x), sin(2x), 1}, V = { solutions to d³y/d x³ +4 dy/dx = 0 }
   3. { cos(2x), sin²(x), 1, cos(2x)+ sin(2x) }, V = { solutions to d³y/d x³ +4 dy/dx = 0 }
   4. { i - 2j + k, i - j - k, i + j - k }, V = R³

4. Let V be a vector space and suppose S = {v₁ ... v_k} spans V. Show that one may discard vectors from S to get a basis for V. Use this to obtain bases for spans of the LD sets in the previous problem.

5. The span of the columns of an m×n matrix A is called the column space of A. Look up an algorithm for finding a basis for the column space; apply it to the matrix A below.

   1	-2	3	3
   2	-5	7	3
   -1	3	-4	3

6. Let U be all functions f in C[0,6] that are linear between the points x = 0, 1, ... 6. (These are linear splines). Sketch the tent function T(x) = max(1 - |x|,0). Show that the set { T(x), T(x-1), ..., T(x-6) } is a basis for U.