The Relationship Betwn Linear Transformations andMatrices 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

andMatrices

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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