Vector arithmetic in Rn
First, some definitions.
For any n = 1, 2, 3, 4, ..., an ordered
n-tuple is a sequence of n real numbers (a1,
a2, ..., an).
The set of all possible n-tuples, denoted by Rn,
is called n-space, and
the n-tuples themselves are called vectors
or n-vectors. The individual
numbers a1, a2,
..., an are called the components
of the vector.
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Operations on n-vectors are defined just as they are for the components of
geometric vectors.
To add or subtract vectors in Rn,
add or subtract their components.
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To multiply a vector in Rn
by a scalar, multiply each component by that scalar. Two vectors are
parallel if one is a scalar
multiple of the other.
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The zero vector 0
in Rn
is the vector with all components equal to 0.
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The negative of any vector
u in Rn
is the vector –u whose components
are the negatives of the components of u.
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Vector operations on Rn
obey the same collection of ten essential rules that geometric vectors do.
| Ten Essential Rules for Vectors in Rn |
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Closure rules:
for vectors u and v
in Rn
and any scalar c,
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u + v
is a vector in Rn
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cu is a vector in Rn
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Arithmetic rules for addition:
for any vectors u, v
and w in Rn,
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u + v
= v + u
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u + (v
+ w) = (u
+ v) + w
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There is a vector 0 such that u
+ 0 = 0
+ u = u
for all vectors u in Rn.
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For each vector u in Rn,
there is a vector –u such that
u + (–u)
= 0.
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Arithmetic rules for scalar multiplication:
for any vectors u and v
in Rn
and any scalars c and d,
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c(u + v)
= cu + cv
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(c + d)u = cu
+ du
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(cd)u = c(du)
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1u = u
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Linear combinations of vectors in Rn
are just extended combinations of additions and scalar multiplications.
A linear combination of
the vectors v1,
v2,
... , vk
in Rn
is any expression of the form
c1v1
+ c2v2
+ ... + ckvk
where c1, c2,
..., ck are all scalars.
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For geometric vectors, we had the basic vectors i
= (1, 0, 0), j = (0, 1, 0) and k
= (0, 0, 1). These vectors are very useful because every geometric vector can
be expressed easily as a linear combination of them. There are n similar vectors
in Rn.
The kth elementary
vector in Rn
is the vector
ek
= (0, 0, ..., 0, 1, 0, ..., 0),
with 1 for the kth component
and 0 for the other components.
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Just as for geometric vectors, every vector in Rn
can be easily expressed as a linear combination of its elementary vectors. In
R5, for
example,
(1, 2, –6, 5, –3) = e1
+ 2e2
– 6e3
+ 5e4
– 3e5.