4
A
REVIEW
OF
VECTOR
AND
MATRIX
ALGEBRA
[PI,
with
the
elements given by:
7
[PI
The multiplication in Equation 1.10 can be written in the abbreviated matrix notation
as
[n/il"l
=
[PI,
(1.11)
Ml
JNJk
=
PIk,
(1.12)
We can
also
use subscripthmmation notation to write
the
same product
as
with
the
implied sum over the
j
index keeping track of the summation. Notice
j
is
in
the second position
of
the
Mtj
term and
the
first position of
the
N,k
term,
so
the summation is over the columns of
[MI
and the rows of
[N],
just as it was in
Equation
1.10.
Equation 1.12
is
an expression for the
iPh
element of the matrix
[PI.
Matrix array notation
is
convenient for doing numerical calculations, especially
when using a computer. When deriving the relationships between the various quan-
tities in physics, however,
matrix
notation is often inadequate because it lacks a
mechanism for keeping track of the geometry of the coordinate system. For example,
in a particular coordinate system, the vector
v
might
be
written as
V
=
lel
+
3e2
+
2C3.
(1.13)
When performing calculations, it is sometimes convenient to use a matrix represen-
tation of this vector by writing:
v+
[V]
=
[;I.
(1.14)
The problem with
this
notation is that there is no convenient way to incorporate the
basis vectors into the matrix.
This
is why we
are
careful to use an arrow
(-)
in
Equation 1.14 instead of an equal sign
(=).
In
this
text, an equal sign between two
quantities means that they
are
perfectly equivalent in every way. One quantity may
be substituted for the other in any expression. For instance, Equation 1.13 implies
that the quantity
1C1
+
3C2
+
2C3
can replace in any mathematical expression, and
vice-versa.
In
contrast,
the
arrow
in
Equation 1.14 implies that
[Vl
can represent
v,
and that calculations can
be
performed using it, but we must
be
careful not to directly
substitute one for the other without specifying the
basis
vectors associated with the
components of
[
Vl
.
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