ma253
,Fa
l
l
2007
–Midterm
1
Solutions
In each of the ten questions that follow, decide whether the statement being
made is true or false. Indicate your decision by writing
T
or
F
in the space provided.
Each question is worth four points.
1.
If a subspace
V
of
R
4
does not contain any of the standard basis vectors
~
e
1
,
~
e
2
,
~
e
3
,
~
e
4
, then it must consist only of the zero vector.
This is
false
. Consider the nextsimplest kind of subspace, the span of one vector,
i.e., a line through the origin in
R
4
. There’s no reason such a line would have to contain
one of the basis vectors.
2.
There exists a system of three linear equations in three unknowns that
has exactly two solutions.
This is
false
, as we discussed in class. The only options for linear spaces are a unique
solution, no solutions, or inFnitely many solutions.
3.
If the vector
~
u
is a linear combination of the vectors
~
v
and
~
w
,th
en
Span
(
~
u,
~
v,
~
w,
~
y
)=
Span
(
~
v,
~
w,
~
y
)
.
This is
true
. We are told that
~
u
=
α
~
v
+
β
~
w
for some
α
and
β
. That allows us
to substitute for
~
u
wherever it appears an expression involving only
~
v
and
~
w
.I
no
t
h
e
r
words, we can rewrite any linear combination of
~
u
,
~
v
,
~
w
and
~
y
as a linear combination
involving only
~
v
,
~
w
and
~
y
. So the spans, i.e., the sets of all possible linear combinations,
are the same.
4.
The only
2
×
2
matrices
A
such that
A
2
=
I
are
±
10
01
²
and
±

0

1
²
.
Very
false
. ±or example,
±

²
,
±
0

1
²
,
±
²
,
±
3
5
4
5
4
5

3
5
²
all have that property. In fact, any re²ection has that property. Just think of what it says
in transformation terms: that doing the transformation twice gives the identity transfor
mation.
5.
If
A
and
B
are
n
×
n
matrices, then
(
A
+
B
)
2
=
A
2
+
2AB
+
B
2
.
This is
false
.O
fcou
r
s
e
,i
f
A
and
B
happen to commute, then it would be true for
those particular matrices
A
and
B
. But it is false in general.
The object of opening the mind, as of opening the mouth, is to shut it again on something solid. – G. K. Chesterton