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2.5 Lines, Planes, and Other Flats
1
Chapter 2.
Dimension, Rank, and Linear
Transformations
2.5
Lines, Planes, and Other Flats
Defnitions 2.4, 2.5.
Let
S
be a subset of
R
n
and let
~a
∈
R
n
.T
h
e
set
{
~x
+
~a

~x
∈
S
}
is the
translate
of
S
by
~a
, and is denoted by
S
+
~a
.
The vector
~a
is the
translation vector
.A
line
in
R
n
is a translate of a
onedimensional subspace of
R
n
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View Full Document2.5 Lines, Planes, and Other Flats
2
Defnition.
If a line
L
in
R
n
contains point (
a
1
,a
2
,...,a
n
) and if vector
~
d
is parallel to
L
,then
~
d
is a
direction vector
for
L
and
~a
=[
a
1
,a
2
,...,a
n
]
is a
translation vector
of
L
.
Note.
With
~
d
as a direction vector and
~a
as a translation vector of a line,
we have
L
=
{
t
~
d
+
~a

t
∈
R
}
.
In this case,
t
is called a
parameter
and
we can express the line
parametrically
as a vector equation:
~x
=
t
~
d
+
~a
or as a collection of component equations:
x
1
=
td
1
+
a
1
x
2
=
td
2
+
a
2
.
.
.
x
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 Fall '04
 IgorDolgachev
 Transformations

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