115A HW 3 SELECTED SOLUTIONS
Sec 1.5.
9.
Let
u
and
v
be distinct vectors in a vector space
V
.
Show that
{
u, v
}
is
linearly dependent if and only if
u
or
v
is a multiple of the other.
If
u
and
v
are linearly dependent, then by definition there exist scalars
a
and
b
,
not both zero, such that
au
+
bv
= 0. If
a
= 0, then
bv
= 0, so
v
= 0, and thus
v
= 0
u
. If
a
= 0, then
u
=

a

1
bv
. In either case,
u
and
v
are multiples of each
other.
Conversely, suppose that
u
is a multiple of
v
. Then
u
=
cv
for some constant
v
,
so
u

cv
= 0. This shows that
{
u, v
}
is linearly dependent.
19. Observe that the problem is equivalent to showing that if
{
A
1
, . . . , A
n
}
is
linearly
dependent
then
{
A
t
1
, . . . , A
t
n
}
is linearly
dependent
.
If
{
A
1
, . . . , A
n
}
is linearly dependent, then there exist scalars
c
i
∈
F
not all zero
such that
c
1
A
1
+
· · ·
+
c
n
A
n
= 0
,
where the 0 on the righthand side of course denotes the zero matrix in
M
n
×
n
(
F
).
By definition of vector addition and scalar multiplication in this vector space (i.e.,
componentwise), this means that
c
1
a
(1)
ij
+
· · ·
+
c
n
a
(
n
)
ij
,
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 Winter '07
 Liu
 Linear Algebra, Algebra, Vectors, Vector Space, linearly independent subset

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