COORDINATES
Math 21b, O. Knill
HOMEWORK: Section 3.4: 2,14,16,22,26,32*,38*
B
COORDINATES. Given a basis
~v
1
, . . .~v
n
, define the matrix
S
=

. . .

~v
1
. . .
~v
n

. . .

.
It is invertible.
If
~x
=
∑
i
c
i
~v
i
, then
c
i
are called the
B
coordinates
of
~v
. We write [
~x
]
B
=
c
1
. . .
c
n
. If
~x
=
x
1
. . .
x
n
, we have
~x
=
S
([
~x
]
B
).
B
coordinates
of
~x
are obtained by applying
S

1
to the coordinates of the standard basis:
[
~x
]
B
=
S

1
(
~x
)
EXAMPLE. If
~v
1
=
1
2
and
~v
2
=
3
5
, then
S
=
1
3
2
5
. A vector
~v
=
6
9
has the coordinates
S

1
~v
=

5
3
2

1
6
9
=

3
3
Indeed, as we can check,

3
~v
1
+ 3
~v
2
=
~v
.
EXAMPLE. Let
V
be the plane
x
+
y

z
= 1. Find a basis, in which every vector in the plane has the form
a
b
0
. SOLUTION. Find a basis, such that two vectors
v
1
, v
2
are in the plane and such that a third vector
v
3
is linearly independent to the first two. Since (1
,
0
,
1)
,
(0
,
1
,
1) are points in the plane and (0
,
0
,
0) is in the
plane, we can choose
~v
1
=
1
0
1
~v
2
=
0
1
1
and
~v
3
=
1
1

1
which is perpendicular to the plane.
EXAMPLE. Find the coordinates of
~v
=
2
3
with respect to the basis
B
=
{
~v
1
=
1
0
,
~v
2
=
1
1
}
.
We have
S
=
1
1
0
1
and
S

1
=
1

1
0
1
. Therefore [
v
]
B
=
S

1
~v
=

1
3
. Indeed

1
~v
1
+ 3
~v
2
=
~v
.
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 Spring '03
 JUDSON
 Math, Linear Algebra, Algebra, Descartes, basis, coordinates, Bcoordinates, basis v1

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