18.06 Problem Set 1 Solution
Due Wednesday, 11 February 2009 at 4 pm in 2106.
Total: 145 points
Problem 1:
If
k
~v
k
= 7 and
k
~w
k
= 3, what are the smallest and largest possible
values of
k
~v
+
~w
k
and
~v
·
~w
?
Solution
(10 points)
k
~v
+
~w
k ≤ k
~v
k
+
k
~w
k
= 10.
k
~v
+
~w
k ≥ k
~v
k  k
~w
k
= 4. (5pts)

~v
·
~w
 ≤ k
~v
k · k
~w
k
= 21. So

21
≤
~v
·
~w
≤
21. (5pts)
The maximum is achieved when the vectors
~v
and
~w
are parallel and pointing to
the same direction, for example,
~v
= (7
,
0
,
0
, . . .
) and
~w
= (3
,
0
,
0
, . . .
); the minimum
is achieved when they are parallel but pointing to opposite directions, for instance,
~v
= (7
,
0
,
0
, . . .
) and
~w
= (

3
,
0
,
0
, . . .
).
REMARK: We should try not to restrict ourselves to the 3dimensional case.
The statement of this problem works for vectors in arbitrary dimensional space.
Problem 2:
Let
A
and
B
be 4
×
4 matrices, and divide each of them into 2
×
2
chunks via
A
=
A
1
A
2
A
3
A
4
and
B
=
B
1
B
2
B
3
B
4
, where
A
1
is the upperleft 2
×
2
corner,
A
2
is the upperright 2
×
2 corner, and so on. Let
C
=
AB
be the 4
×
4
product of
A
and
B
, and similarly divide
C
into 2
×
2 chunks as
C
=
C
1
C
2
C
3
C
4
.
(a) Give formulas for these 2
×
2 chunks
C
1
...
4
in terms of matrix products and
sums of the chunks
A
1
...
4
and
B
1
...
4
(your final formulas should
not
reference
the individual numbers within those chunks).
(b) Justify your formulas by an example (come up with 4
×
4 matrices
A
and
B
with nonzero entries, multiply them to get
C
, and compare to your formulas
in terms of 2
×
2 chunks—it is acceptable to check just one of the output 2
×
2
chunks, say
C
2
).
Solution
(15 points = 10+5)
(a) Similar to usual matrix multiplication, matrix multiplication by blocks for
mally has the same form.
C
1
=
A
1
B
1
+
A
2
B
3
,
C
2
=
A
1
B
2
+
A
2
B
4
,
C
3
=
A
3
B
1
+
A
4
B
3
,
C
4
=
A
3
B
2
+
A
4
B
4
.
1