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The Relationship Betwn Linear Transformations andMatrices 203
Because of Fact 4, we know that both and have
the same rank.
One look at the simplified matrix is enough to see that only and
are linearly independent among its columns.
This means it has a rank of 2, and so does our initial matrix.
1
0
0
1
3
1
1
2
1
0
0
1
0
0
0
0
1
0
0
1
The Relationship Between Linear Transformations
andMatrices
We talked a bit about the relationship between linear transformations and
matrices on page 168. We said that a linear transformation from R
n
to R
m
could be written as an m×n matrix:
a
12
a
22
a
m2
a
1n
a
2n
a
mn
a
11
a
21
a
m1
As you probably noticed, this explanation is a bit vague. The more exact rela-
tionship is as follows:
The relationship between linear Transformations and matrices
If is an arbitrary element in R
n
and f is a function from R
n
to R
m
,
then f is a linear transformation from R
n
to R
m
if and only if
for some matrix A.
x
1
x
2
x
n
=
x
1
x
2
x
n
x
1
x
2
x
n
a
12
a
22
a
m2
a
1n
a
2n
a
mn
a
11
a
21
a
m1
f

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