We can often visualize transformations between euclidean spaces. For example, the name of
transformation `Reflection in x-axis`: **R**^{2} → **R**^{2},
(`x`, `y`) → (`x`, -`y`) is inspired by the picture.

Example The transformation
`Reflection in y-axis`: **R**^{2} → **R**^{2},
(`x`, `y`) → (-`x`, `y`)

is given by the matrix

[ | -1 | 0 | ] |

0 | 1 |

The transformation `Rotation by &\theta;`: **R**^{2} → **R**^{2},
(`x`, `y`) → (`x`cos`θ` + `y`sin`θ`, - `x`sin`θ` + `y`cos`θ`)

is given by the matrix (remember writing vectors vertically)

[ | cosθ |
- sinθ |
] |

sinθ |
cosθ |

The transformation `Projection to (x`_{1}, `x`_{3}`)-plane`:
**R**^{3} → **R**^{2},
(`x`_{1}, `x`_{2}, `x`_{3}) → (`x`_{1}, `x`_{3})

is given by the matrix

[ | 1 | 0 | 0 | ] |

0 | 0 | 1 |

Linear transformations are characterized by the property that addition and scalar multiplication are preserved. Geometrically, this means that parallelograms and stretching/shrinking are preserved.

Example The transformation
`Reflection in x-axis` clearly preserves parallelograms and stretching/shrinking.

The rotation transformation also preserves the two operations.

The projection transformation is also a linear transformation.