# Linear Algebra/OLD/Change of Basis

## Change of Basis

[edit | edit source]It was shown earlier that a square matrix can represent a linear transformation of a vector space into itself, and that this matrix is dependent on the basis chosen for the vector space. We will now show how to change the basis of a given vector space

Suppose we have some vector space whose basis is given by the set and we would like to change it to the set . The basis * B* still belongs to vector space aforementioned, so its vectors can be expressed as a linear combination [

*Eq. 1*]

,

,

Each vector of the set * B* has a coordinate matrix with respect to the basis we started off with, namely the set designated as

*. We represent this as*

**A**

Setting these coordinate matrices as the columns of a matrix P gives us a **transition matrix**. This transition matrix transforms the original basis * A* to a new basis

*of some vector space. The transition matrix is actually the transpose of [*

**B***Eq. 1*] that we saw earlier

To summarize, in order to find the transition matrix from some basis * F* to some basis

*, we must compute the coordinate vector for each element of our original basis*

**G***with respect to the other basis*

**F***. The matrix whose columns are formed by the coordinate vectors is the transition matrix.*

**G**## Proof

[edit | edit source]*Theorem 1*

[edit | edit source]*If ***P*** is the transition matrix from the basis A to the basis Z, and *

**β**

*is an element of the vector space, then it follows that*

#### Proof

[edit | edit source]