We have seen matrix transformations are linear. Conversely, this example suggests the converse is also true.

Any linear transformation ` T`:

` A` = [

In other words, the columns of the matrix ` A` are the values of the linear transformation

Example
The identity transformation ** id**:

[(ide_{1})
(ide_{2}) ...
(id)]
= [e_{n}e_{1} e_{2} ... ]
= [e_{n} |
1 | 0 | . . . | 0 | ] |

0 | 1 | . . . | 0 | ||

: | : | : | |||

0 | 0 | . . . | 1 |

called the identity matrix and denoted by **I**_{n}.

Example In this example, we started from a matrix

= [A |
1 | -1 | 1 | ] |

3 | 1 | -1 | ||

2 | 2 | 1 |

and produced a matrix transformation (which is also a linear transformation).

[T |
x_{1} |
] = [ | x_{1} - x_{2} + x_{3} |
] : R^{3} → R^{3} |

x_{2} |
3x_{1} + x_{2} - x_{3} |
|||

x_{3} |
2x_{1} + 2x_{2} + x_{3} |

Conversely, we may recover the columns of ` A` as follows

[col 1] of = A[T |
1 | ] = [ | 1 - 0 + 0 | ] = [ | 1 | ] |

0 | 3 + 0 - 0 | 3 | ||||

0 | 2 + 0 + 0 | 2 |

[col 2] of = A[T |
0 | ] = [ | 0 - 1 + 0 | ] = [ | -1 | ] |

1 | 0 + 1 - 0 | 1 | ||||

0 | 0 + 2 + 0 | 2 |

[col 3] of = A[T |
0 | ] = [ | 0 - 0 + 1 | ] = [ | 1 | ] |

0 | 0 + 0 - 1 | -1 | ||||

1 | 0 + 0 + 1 | 1 |

Example The reflections and rotations on **R**^{2} are linear transformations
because they preserve parallelogram and stretching/shrinking.
To find their matrices, we simply need to apply reflections and rotations to the standard basis.

The picture shows that the reflections of **e**_{1}, **e**_{2}
in `x`-axis are **e**_{1} = (1, 0), -**e**_{2} = (0, -1).
Therefore the matrix for the reflection is

[ | 1 | 0 | ] |

0 | -1 |

Moreover, we also see from picture that for the rotation by angle θ,
` T`(

Therefore the matrix for the rotation is

[ | cosθ |
- sinθ |
] |

sinθ |
cosθ |

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[Extra: Matrix of a linear transformation]