### Change of Basis

##### 1. Matrix for change of coordinates

Let `B` = {`b`_{1}, `b`_{2}, `b`_{3}} and `C` = {`c`_{1}, `c`_{2}, `c`_{3}} be two bases of a vector space `V`. Then the vectors in `B` may be written as linear combinations of vectors in `C`.

`b`_{1} = `a`_{11}**c**_{1} + `a`_{12}**c**_{2} + `a`_{13}**c**_{3}

`b`_{2} = `a`_{21}**c**_{1} + `a`_{22}**c**_{2} + `a`_{23}**c**_{3}

`b`_{3} = `a`_{31}**c**_{1} + `a`_{32}**c**_{2} + `a`_{33}**c**_{3}

The matrix

[`B`]_{C} = [ [`b`_{1}]_{C} [`b`_{2}]_{C} [`b`_{3}]_{C} ] = [ |
`a`_{11} |
`a`_{21} |
`a`_{31} |
], |

`a`_{12} |
`a`_{22} |
`a`_{32} |

`a`_{13} |
`a`_{23} |
`a`_{33} |

with a very suggestive notation, is the "coordinates of `B` with respect to `C`". The next result suggests that [`B`]_{C} is the matrix for changing the coordinates in `B` to the coordinates in `C`.

The coordinates of **v** ∈ **V** in the bases `B` and `C` are related by

[`v`]_{C} = [`B`]_{C}[`v`]_{B}.

Proof Without loss of generality, we may assume `V` = **R**^{3}. We use the bases vectors as columns and construct invertible matrices [`B`] = [`b`_{1} `b`_{2} `b`_{3}], [`C`] = [`c`_{1} `c`_{2} `c`_{3}]. Then

[`B`] = [`a`_{11}**c**_{1} + `a`_{12}**c**_{2} + `a`_{13}**c**_{3} `a`_{21}**c**_{1} + `a`_{22}**c**_{2} + `a`_{23}**c**_{3} `a`_{31}**c**_{1} + `a`_{32}**c**_{2} + `a`_{33}**c**_{3}]

= [`c`_{1} `c`_{2} `c`_{3}] [ |
`a`_{11} |
`a`_{21} |
`a`_{31} |
] = [`C`][`B`]_{C}. |

`a`_{12} |
`a`_{22} |
`a`_{32} |

`a`_{13} |
`a`_{23} |
`a`_{33} |

Let the coordinates of `v` ∈ `V` to be [`v`]_{B} = (`x`_{1}, `x`_{2}, `x`_{3}) and [`v`]_{C} = (`y`_{1}, `y`_{2}, `y`_{3}). Then

**v** = `x`_{1}`b`_{1} + `x`_{2}`b`_{2} + `x`_{3}`b`_{3} = [`B`][`v`]_{B} = [`C`][`B`]_{C}[`v`]_{B}.

Compared with

**v** = `y`_{1}`c`_{1} + `y`_{2}`c`_{2} + `y`_{3}`c`_{3} = [`C`][`v`]_{C},

we have [`B`]_{C}[`v`]_{B} = [`v`]_{C}.

We list some properties of the matrix for changing the coordinates.

- transitivity
- [
`C`]_{D} [`B`]_{C} = [`B`]_{D}
- inverse
- [
`B`]_{C}^{-1} = [`C`]_{B}
- identity
- [
`B`]_{B} = `I`

The transitivity property means that changing the `B`-coordinates to the `C`-coordinates and then further changing to the `D`-coordinates is the same as changing the `B`-coordinates to the `D`-coordinates. The identity property means that the change from the `B`-coordinates to the `B`-coordinates is no change at all. The inverse follows from the transitivity property (take `D` = `B`) and the identity property.