University of Colorado, Department of Physics
PHYS3220, Fall 09, HW#12
due Wed, Nov 18, 2PM at start of class
1. Vectors (Total: 20 pts)
a) (Griffiths, Problem A.1) Consider the ordinary vectors in 3D (
a
x
ˆ
i
+
a
y
ˆ
j
+
a
z
ˆ
k
), with
complex components. For each of the following three subsets find out whether or not
it constitutes a vector space. If so, what is the dimension of the vector space? If not,
why is it not a vector space?
(i) The subset of all vectors with
a
z
= 0.
(ii) The subset of all vectors whose
z
component is 1
(iii) The subset of all vectors whose components are all equal.
b) Does the subset of all 2
×
2 matrices form a vector space? Assume the usual rules for
matrix addition and multiplication by a scalar, namely:
a
b
c
d
¶
+
e
f
g
h
¶
=
a
+
e
b
+
f
c
+
g
d
+
h
¶
,
α
a
b
c
d
¶
=
αa
αb
αc
αd
¶
If it does not form a vector space, why not? If it does form a vector space, state the
dimensionality and give an example of a set of basis vectors.
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 Fall '08
 STEVEPOLLOCK
 Physics, Vector Space, Hilbert space, Hermitian, hermitian adjoint, Hermitian adjoint operator

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