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Lecture 6
Generating Sets and Bases
Let
V
be the vector space
R
2
and consider the vectors (1
,
0)
,
(0
,
1). Then,
every vector (
x,y
)
∈
R
2
can be written as a combination of those vectors.
That is:
(
x,y
) =
x
(1
,
0) +
y
(0
,
1)
.
Similarly, the two vectors (1
,
1) and (1
,
2) do not belong to the same line,
and every vector in
R
2
can be written as a combination of those two vectors.
6.1 Introduction
In particular:
(
x,y
) =
a
(1
,
1) + (1
,
2)
gives us two equations
a
+
b
=
x
and
a
+ 2
b
=
y
Thus, by substituting the ﬁrst equation to the second, we get
b
=

x
+
y
Inserting this into the ﬁrst equation we get
a
= 2
x

y
Take for example the point (4
,
3). Then:
(4
,
3) = 5(1
,
1) + (

1)(1
,
2)
= 5(1
,
1)

(1
,
2)
35
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LECTURE 6. GENERATING SETS AND BASES
We have similar situation for
R
3
and all of the spaces
R
n
.
In the case of
R
3
, for example, every vector can be written as combinations
of (1
,
0
,
0)
,
(0
,
1
,
0) and (0
,
0
,
1), i.e.,
(
x,y,z
) =
x
(1
,
0
,
0) +
y
(0
,
1
,
0) +
z
(0
,
0
,
1)
.
Or, as a combination of (1
,

1
,
0)
,
(1
,
1
,
1) and (0
,
1
,

1), that is:
(
x,y,z
) =
a
(1
,

1
,
0) +
b
(1
,
1
,
1) +
c
(0
,
1
,

1)
.
The latter gives three equation:
a
+
b
=
x
(1)

a
+
b
+
c
=
y
(2)
b

c
=
z
(3)
.
(2) + (3) gives:

a
+ 2
b
=
y
+
z
(4)
(4) + (1) gives:
3
b
=
x
+
y
+
z
or
b
=
x
+
y
+
z
3
.
Then (1) gives:
a
=
x

b
=
x

x
+
y
+
z
3
=
2
x

y

z
3
.
Finally, (3) gives:
c
=
b

z
=
x
+
y

2
z
3
Hence, we get:
(
x,y,z
) =
2
x

y

z
3
(1
,

1
,
0) +
x
+
y
+
z
3
(1
,
1
,
1) +
x
+
y

2
z
3
(0
,
1
,

1)
.
6.2. IN GENERAL
37
6.2 In general
Notice that we get only one solution, so there is only one
way that we can write a
vector in
R
3
as a combination of those vectors. In general, if we have
k
vectors in
R
3
, then the equation:
x
= (
x
1
,x
2
,...,x
n
) =
c
1
v
1
+
c
2
v
2
+
...
+
c
k
v
k
(
*
)
gives
n
equations involving the
n
coordinates of
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This note was uploaded on 11/23/2011 for the course MATH 2025 taught by Professor Staff during the Spring '08 term at LSU.
 Spring '08
 Staff
 Vectors, Vector Space, Sets

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