A linear transformation (also called a linear mapping ) is a transformation such that
T
:
R
n
→
R
m
{\displaystyle {\rm T}:\R^n\to\R^m}
T
(
a
→
+
b
→
)
=
T
(
a
→
)
+
T
(
b
→
)
{\displaystyle {\rm T}(\vec{a}+\vec{b})={\rm T}(\vec{a})+{\rm T}(\vec{b})}
T
(
c
a
→
)
=
c
T
(
a
→
)
{\displaystyle {\rm T}(c\vec{a})=c{\rm T}(\vec{a})}
If a transformation is linear, there will be an associated transformation matrix .
Examples [ ] The transformation
T
[
x
1
x
2
]
=
[
x
2
3
x
2
]
{\displaystyle {\rm T}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}x_2\\3x_2\end{bmatrix}}
is linear because
T
[
c
x
1
c
x
2
]
=
[
c
x
2
3
c
x
2
]
=
c
[
x
2
3
x
2
]
=
c
T
[
x
1
x
2
]
{\displaystyle {\rm T}\begin{bmatrix}cx_1\\cx_2\end{bmatrix}=\begin{bmatrix}cx_2\\3cx_2\end{bmatrix}
=c\begin{bmatrix}x_2\\3x_2\end{bmatrix}=c{\rm T}\begin{bmatrix}x_1\\x_2\end{bmatrix}}
and
T
[
a
+
b
c
+
d
]
=
[
c
+
d
3
c
+
3
d
]
=
[
c
3
c
]
+
[
d
3
d
]
=
T
[
a
c
]
+
T
[
b
d
]
{\displaystyle {\rm T}\begin{bmatrix}a+b\\c+d\end{bmatrix}=\begin{bmatrix}c+d\\3c+3d\end{bmatrix}
=\begin{bmatrix}c\\3c\end{bmatrix}+\begin{bmatrix}d\\3d\end{bmatrix}={\rm T}\begin{bmatrix}a\\c\end{bmatrix}+{\rm T}\begin{bmatrix}b\\d\end{bmatrix}}
The transformation
T
[
x
1
x
2
]
=
[
x
1
x
2
5
]
{\displaystyle {\rm T}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}x_1x_2\\5\end{bmatrix}}
is not linear because
T
[
c
x
1
c
x
2
]
=
[
c
x
1
c
x
2
5
]
=
[
c
2
x
1
x
2
5
]
≠
c
T
[
x
1
x
2
]
=
c
[
x
1
x
2
5
]
{\displaystyle {\rm T}\begin{bmatrix}cx_1\\cx_2\end{bmatrix}=\begin{bmatrix}cx_1cx_2\\5\end{bmatrix}
=\begin{bmatrix}c^2x_1x_2\\5\end{bmatrix}\ne c{\rm T}\begin{bmatrix}x_1\\x_2\end{bmatrix}=c\begin{bmatrix}x_1x_2\\5\end{bmatrix}}
In general, if any variable is raised to a power or two variables are multiplied by the transformation, or if there are any constants other than 0 as elements of the transformed vector, the transformation will not be linear.
