Mathematic 104, Fall 2010: Assignment #1
Due:
Wednesday, October 6th
Instructions:
Please ensure that your answers are legible. Also make sure that all steps are shown – even
for problems consisting of a numerical answer. Bonus problems cover advanced material and, while good
practice, are
not
required and will
not
be graded.
Problem #1.
Consider the following 4 vectors:
v
1
=
2
1
1
,v
2
=
1
1
0
,v
3
=
5
2
3
,v
4
=
10
6
4
Let
E
=
Span
{
v
1
,v
2
,v
3
,v
4
}
a) Can the
v
j
form a basis for
E
? Please justify your answer.
b) Determine
dim
(
E
).
c) Write down a matrix whose null space is
E
.
d) Find a vector
w
so that
{
v
1
,v
2
,w
}
form a basis of
C
3
.
Problem #2.
Let
A
and
B
be 2
×
2 matrices.
a) Find
A
and
B
so that
AB
6
=
BA
.
b) Now ﬁx
A
, and suppose that we know that
AB
=
BA
for every 2
×
2 matrix
B
. Show that
A
must be
a multiple of the identity matrix i.e. of the form
A
=
±
a
0
0
a
²
=
aI.
Problem #3.

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