Page 196 - Matrix Analysis & Applied Linear Algebra

4.3 Linear Independence 191

4.3.3. Suppose that in a population of a million children the height of each one
is measured at ages 1 year, 2 years, and 3 years, and accumulate this
data in a matrix

1 yr 2 yr 3 yr
 
#1 h 11 h 12 h 13
#2  h 21 h 22 h 23 

.  . .

.  . . . 
. . . . .  = H.
 
#i  h i1 h i2 h i3 
.  . . . 
. . . .
. . . .
Explain why there are at most three “independent children” in the sense
that the heights of all the other children must be a combination of these
“independent” ones.

4.3.4. Consider a particular species of wildflower in which each plant has several
stems, leaves, and flowers, and for each plant let the following hold.
S = the average stem length (in inches).
L = the average leaf width (in inches).
F = the number of flowers.
Four particular plants are examined, and the information is tabulated
in the following matrix:

S L F
 
#1 1 1 10
#2  2 1 12 
A =   .
#3  2 2 15 
#4 3 2 17
For these four plants, determine whether or not there exists a linear rela-
tionship between S,L, and F. In other words, do there exist constants
α 0 ,α 1 ,α 2 , and α 3 such that α 0 + α 1 S + α 2 L + α 3 F =0 ?

4.3.5. Let S = {0} be the set containing only the zero vector.
(a) Explain why S must be linearly dependent.
(b) Explain why any set containing a zero vector must be linearly
dependent.

4.3.6. If T is a triangular matrix in which each t ii =0, explain why the rows
and columns of T must each be linearly independent sets.