11-coordinates - COORDINATES HOMEWORK Section 3.4...

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 .