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The vector V in Equation (2.1) may also be represented by a matrix having a
single column and n rows:
V
V

 
 
1
2
V
=  
 
.
(2.2)
V
n
We treat this column vector as having a meaning identical to that of the comma-
separated list of components written in Equation (2.1). Vectors are normally ex-
pressed in these forms, but we sometimes need to express vectors as a matrix
consisting of a single row and n columns. We write row vectors as the transpose
of their corresponding column vectors:
]
T
[
V
=
VV
.
V
(2.3)
1
2
n
A vector may be multiplied by a scalar to produce a new vector whose com-
ponents retain the same relative proportions. The product of a scalar a and a vec-
tor V is defined as
a
VV
==
a
aV
,
aV
,
.
,
aV
(2.4)
1
2
n
In the case that
a
=−
1
, we use the slightly simplified notation
V to represent the
negation of the vector V .
Vectors add and subtract componentwise. Thus, given two vectors P and Q ,
we define the sum
PQ as
+
PQ
+= +
PQP Q
,
+
,
,
P Q
+
.
(2.5)
1
1
2
2
n
n
The difference between two vectors, written
Q , is really just a notational sim-
P
(
)
plification of the sum
PQ .
With the above definitions in hand, we are now ready to examine some fun-
damental properties of vector arithmetic.
+−
Theorem 2.1. Given any two scalars a and b , and any three vectors P , Q , and
R , the following properties hold.
(a)
P
+=+
QQP
(
)
(
)
(b)
PQ R P QR
++=++
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