MATH 223, Linear Algebra
Winter, 2010
Assignment 1, due
in class
January 18, 2010
1. Let
z
= 5 + 3
i
and
w
= 4

i
. Find ¯
z
, ¯
w
,
z
+
w
,
z

w
,
z
·
w
and
z
w
(all
in the form
a
+
bi
with
a
and
b
real numbers). Find the absolute value of
each of these 6 numbers.
2.
(a) Prove that for any complex numbers
z
1
and
z
2
,
z
1
+
z
2
=
z
1
+
z
2
and
z
1
·
z
2
=
z
1
·
z
2
You may use any properties about multiplication and
addition of real numbers.
(b) If
A
is a matrix over the complex numbers, we let
¯
A
be the most
obvious thing — it is obtained from
A
by replacing each entry by its
conjugate. Supposing that
A
·
B
is defined, show that
A
·
B
=
¯
A
·
¯
B
.
3. Solve each of the following systems of equations. That is, find the unique
solution if there is one, the general solution in vector parametric form if
there is more than one solution, or explain why there is no solution if that
is the case. Use augmented matrices.
(a) This one’s over the field
R
, the reals.
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 Winter '10
 Loveys
 Linear Algebra, Real Numbers, Complex Numbers, Complex number

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