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Multiplication of Vectors by scalars
(Scalar Multiplication):
R
×
R
n
→
R
n
(
c,
x
)
7→
c
x
‡
c,
x
1
x
2
.
.
x
n
·
7→
cx
1
cx
2
.
.
cx
n
Addition of Vectors:
R
n
×
R
n
→
R
n
(
x
,
y
)
7→
x
+
y
‡
x
1
x
2
.
.
x
n
,
y
1
y
2
.
.
y
n
·
7→
x
1
+
y
1
x
2
+
y
2
.
.
x
n
+
y
n
Properties of these Operations: see Theorem 1.3 page 10.
These properties are consequences of the fact that in each coordi
nate we multiply or add real numbers. These operations inherit of the
properties of addition and multiplication of real numbers.
0
is neutral (rather than identity) for addition,

x
is the opposite
of
x
(rather than inverse for addition).
These properties give sense to Linear Combinations.
Exercises to make sure you understand the deﬁnitions:
7
,
8
,
6
.
1
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View Full DocumentDot (Inner) Product:
R
n
×
R
n
→
R
(
x
,
y
)
7→
x
.
y
‡
x
1
x
2
.
.
x
n
,
y
1
y
2
.
.
y
n
·
7→
x
1
y
1
+
...
+
x
n
y
n
Properties of the Dot Product: see Theorem 1.5 page 17.
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 Spring '08
 HIETMANN
 Addition, Multiplication, Vectors, Scalar

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