Math 603-601 - Fall 2002
Homework
Assignment 2
- 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.
- B={1, 2x, 4x2-1} and D={1, x, x2}, where
v is the polynomial p(x)=x(3-2x).
- 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.
- Let B = {v1 ... vn}, D =
{w1 ... wn}, and E =
{u1 ... un} be bases for a vector
space V. Suppose that you are given these column matrices:
A1 = [ [v1]E ...
[vn]E]
A2 = [ [w1]E ...
[wn]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.
-
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}.
- 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.
-
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)
-
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.)
Due Thursday, 19 Sept.