Page 272 - A Course in Linear Algebra with Applications

256 Chapter Seven: Orthogonality in Vector Spaces

' xi + x 2 - 2x 3 — 3
^ I 2xi - x 2 + 3x 3 = 4
1
| xi + x 3 = 1
. X\ + X 2 + X 3 = 1

2. The following data were collected for the mean annual tem-
perature t and rainfall r in a certain region; use the Method
of Least Squares to find a linear approximation for r in terms
of t (a calculator is necessary):

t 24 27 22 24
r 47 30 35 38

3. In a tropical rain forest the following data was collected for
the numbers x and y (per square kilometer) of a prey species
and a predator species over a number of years. Use least
squares to find a quadratic function of x that approximates y
(a calculator is necessary):

X 2 3 4 5
y l 2 2 1

4. Find the optimal least squares solution of the linear system

Xi + 2x 3 = 1
+ = 0
x 2 3x 3
—xi + x 2 + x 3 = 0
— x 2 — 3x 3 — 1

5. Find a least squares approximation to the function e~ x by
a linear function in the interval [1, 2]. [Use the inner product
< f,9> = fi f(x)g(x)dx}.
6. Find a least squares approximation for the function sin x
as a quadratic function of x in the interval [0, n]. [Here the
inner product < f,g > = JQ f(x)g(x)dx is to be used].