# What is Transformation: Definition and 1000 Discussions

In linear algebra, linear transformations can be represented by matrices. If

T

{\displaystyle T}
is a linear transformation mapping

R

n

{\displaystyle \mathbb {R} ^{n}}
to

R

m

{\displaystyle \mathbb {R} ^{m}}
and

x

{\displaystyle \mathbf {x} }
is a column vector with

n

{\displaystyle n}
entries, then

T
(

x

)
=
A

x

{\displaystyle T(\mathbf {x} )=A\mathbf {x} }
for some

m
×
n

{\displaystyle m\times n}
matrix

A

{\displaystyle A}
, called the transformation matrix of

T

{\displaystyle T}
. Note that

A

{\displaystyle A}
has

m

{\displaystyle m}
rows and

n

{\displaystyle n}
columns, whereas the transformation

T

{\displaystyle T}
is from

R

n

{\displaystyle \mathbb {R} ^{n}}
to

R

m

{\displaystyle \mathbb {R} ^{m}}
. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.

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