Physics 570
Homework No. 1
due Wednesday, 27 January, 2010
1.
Consider a standard 2dimensional vector space, with the usual Cartesian basis set,
{
ˆ
x,
ˆ
y
}
, and
a particular vector
~
B
≡
3ˆ
x

2ˆ
y
. Now consider a diﬀerent pair of basis vectors,
e
1
≡
2ˆ
x

ˆ
y ,
e
2
≡
ˆ
x
+ ˆ
y .
Please determine “the components” of the vector
~
B
relative to this basis. However, I want to give
you two distinct deﬁnitions of the word components. The purpose of the components of a vector
is to determine uniquely the vector in question. Both of these deﬁnitions serve that purpose.
Therefore, I ask that you determine both of them, and then the relation between them.
a.
The ﬁrst deﬁnition, for the components
{
B
i
}
2
1
, is given by the following equality:
~
B
=
B
1
e
1
+
B
2
e
2
.
b.
The second deﬁnition, for the components
{
B
j
}
2
1
, is given by the following deﬁnitions:
B
1
≡
~
B
·
e
1
,
B
2
≡
~
B
·
e
2
,
where the “dot product” used is the usual, Cartesian dot product that we always use.
c.
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 Spring '10
 DavidS.King
 Linear Algebra, Vector Space, General Relativity, Special Relativity

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