Math 603-601 - Fall 2002
Homework
Assignment 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.)
- Let Pn be polynomials of degree n or less in x.
- Show that P2 is a subspace of P3
- Let U be the subset of Pn 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?
- Determine whether the following sets of vectors are LI or LD.
- { 1-x, 1+2x,1+x}, V = P1
- { cos(2x), sin(2x), 1}, V = { solutions to d3y/d
x3 +4 dy/dx = 0 }
- { cos(2x), sin2(x), 1, cos(2x)+ sin(2x) }, V = {
solutions to d3y/d x3 +4 dy/dx = 0 }
- { i - 2j + k, i - j -
k, i + j - k }, V = R3
- Let V be a vector space and suppose S = {v1 ...
vk} 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.
- 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 |
- 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.
Due Thursday, 12 Sept.