2.035:
Midterm
Exam
 Part
1
Spring
2007
SOLUTION
PROBLEM
1:
a)
A
vector
space
is
a
set
V
of
elements
called
vectors
together
with
operations
of
addition
and
multiplication
by
a
scalar,
where
these
operations
must
have
the
following
properties:
(A)
Corresponding
to
every
pair
of
vectors
x
,
y
∈
V
there
is
a
vector
in
V
,
denoted
by
x
+
y
,
and
called
the
sum
of
x
and
y
,
with
the
following
properties:
(1)
x
+
y
=
y
+
x
for
all
x
,
y
∈
V
;
(2)
x
+ (
y
+
z
) = (
x
+
y
) +
z
for
all
x
,
y
,
z
∈
V
;
(3)
there
is
a
unique
vector
in
V
,
denoted
by
o
and
called
the
null
vector,
with
the
property
that
x
+
o
=
x
for
all
x
∈
V
;
and
(4)
corresponding
to
every
vector
x
∈
V
there
is
a
unique
vector
in
V
,
denoted
by
−
x
with
the
property
that
x
+ (
−
x
) =
o
.
(B)
Corresponding
to
every
real
number
α
∈
R
and
every
vector
x
∈
V
there
is
a
vector
in
V
,
denoted
by
α
x
,
and
called
the
product
of
α
and
x
,
with
the
following
properties:
(5)
α
(
β
x
) = (
αβ
)
x
for
all
α, β
∈
R
and
all
x
∈
V
;
(6)
α
(
x
+
y
) =
α
x
+
α
y
for
all
α
∈
R
and
all
x
,
y
∈
V
;
(7)
(
α
+
β
)
x
=
α
x
+
β
x
for
all
α, β
∈
R
and
all
x
∈
V
;
and
(8)
1
x
=
x
for
all
x
∈
V
.
b)
A
set
of
vectors
{
f
1
,
f
2
, . . . ,
f
n
}
is
said
to
be
linearly
independent
if
the
only
scalars
α
1
, α
2
, . . . , α
n
for
which
α
1
f
1
+
α
2
f
2
. . .
+
α
n
f
n
=
o
are
α
1
=
α
2
=
. . .
=
α
n
=
0.
c)
If
a
vector
space
V
contains
a
linearly
independent
set
of
n
(
>
0)
vectors
but
contains
no
linearly
independent
set
of
n
+
1
vectors
we
say
that
the
dimension
of
V
is
n
.
d)
If
V
is
a
n
dimensional
vector
space
then
any
set
of
n
linearly
independent
vectors
is
called
a
basis
for
V
.
e)
If
{
f
1
,
f
2
, . . . ,
f
n
}
is
a
basis
for
an
n
dimensional
vector
space
V
,
then
any
vector
x
∈
V
can
be
expressed
in
the
form
x
=
ξ
1
f
1
+
ξ
2
f
2
+
. . .
+
ξ
n
f
n
where
the
set
of
scalars
ξ
1
, ξ
2
, . . . , ξ
n
is
unique
and
are
called
the
components
of
x
in
the
basis
{
f
1
,
f
2
, . . . ,
f
n
}
.
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f)
To
every
pair
of
vectors
x
,
y
∈
V
we
associate
a
real
number
denoted
by
x
y
and
called
the
·
scalar
product
of
x
and
y
provided
that
this
product
has
the
following
properties:
(9)
x
y
=
y
x
for
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 Spring '07
 RohanAbeyaratne
 Linear Algebra, Vector Space, det A.

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