In other words,
t
6
= 1.
(b) Are the vectors
a
1
and
a
2
from part (a) linearly independent? Explain your
answer.
(c) Let
a
1
,
a
2
and
a
3
be as in (a). Let
a
4
=

5
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Chapter 10 / Exercise 1
Mathematical Applications for the Management, Life, and Social Sciences
Harshbarger
Expert Verified
(5) (after 2.3) (Geometry Question) Consider the infinite system of linear equations in
two variables given by
ax
+
by
= 0 where (
a, b
) moves along the unit circle in the
plane.
(a) How many solutions does this system have?
(b) How many linearly independent equations in the above system give you the
same set of solutions? Write down two separate such linear systems, in vector form.
(c) What happens to the infinite linear system if you add to it the equation
0
x
+ 0
y
= 0?
(d) What happens to the infinite linear system if one of the equations slightly
perturbs to
ax
+
by
=
c
where
c
is a small positive number?
(6) For each of the situations described below,
give an example
(if it’s possible) or
explain why it’s not possible
.
(a) A set of vectors that does not span
R
3
. After adding one more vector, the set
does span
R
3
.
1
(b) A set of vectors that are linearly dependent. After adding one more vector, the