Transformations

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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