This implies that
x
can also be expressed as a linear combination of elements in
{
x
1
, 𝑥
2
, ...,
x
𝑘
}
, that is there exist scalars
𝛼
1
, 𝛼
2
, . . . , 𝛼
𝑘
such that
x
=
𝑘
∑
𝑖
=1
𝛼
𝑖
x
𝑖
or
x
=
𝑘
∑
𝑖
=1
𝛼
𝑖
x
𝑖
=
𝑛
∑
𝑖
=
𝑘
+1
𝛼
𝑖
x
𝑖
which contradicts the assumption that
𝐵
is a basis for
𝑋
. Therefore
{
𝑓
(
x
𝑘
+1
)
, . . . , 𝑓
(
x
𝑛
)
}
is a basis for
𝑓
(
𝑋
) and therefore dim
𝑓
(
𝑥
) =
𝑛
−
𝑘
. We conclude that
dim kernel
𝑓
+ dim
𝑓
(
𝑋
) =
𝑛
= dim
𝑋
3.25
Equation (3.2) implies that nullity
𝑓
= 0, and therefore
𝑓
is onetoone (Exercise
3.18).
3.26
Choose some
x
= (
𝑥
1
, 𝑥
2
, . . . , 𝑥
𝑛
)
∈
𝑋
.
x
has a unique representation in terms of
the standard basis (Example 1.79)
x
=
𝑛
∑
𝑗
=1
𝑥
𝑗
e
𝑗
Let
y
=
𝑓
(
x
). Since
𝑓
is linear
y
=
𝑓
(
x
) =
𝑓
⎛
⎝
𝑛
∑
𝑗
=1
𝑥
𝑗
e
𝑗
⎞
⎠
=
𝑛
∑
𝑗
=1
x
𝑗
𝑓
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 Fall '10
 Dr.DuMond
 Macroeconomics, Linear Algebra, basis, Michael Carter, unique representation, Foundations of Mathematical Economics

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