Change of Basis

A linear transformation can map a vector space onto itself and change it’s basis. This is known as a change of basis.

Since every linear transformation can be expressed as a matrix, the matrix that performs this change of basis is known as a transition matrix.

Definition §

Transition Matrix

Let $\mathcal{B},\mathcal{C}$ be two different bases for the vector field $V$. Then if $\vec{v}\in V$

$$[\vec{v}{]}_{\mathcal{C}}=P_{\mathcal{C},\mathcal{B}}\cdot [\vec{v}{]}_{\mathcal{B}}\forall \vec{v}$$

Then there exists a unique matrix $P_{\mathcal{C},\mathcal{B}}$, which is the transition matrix from $\mathcal{B}$ to $\mathcal{C}$, and is given by:

$$P_{\mathcal{C},\mathcal{B}}=\left[\begin{array}{cccc}[{\vec{b}}_1{]}_{\mathcal{C}}& [{\vec{b}}_2{]}_{\mathcal{C}}& \dots & [{\vec{b}}_n{]}_{\mathcal{C}}\end{array}\right]$$

where $\mathcal{B}=\{{\vec{b}}_1,{\vec{b}}_2,\dots ,{\vec{b}}_n\}$

Chaining Transition Matrices §

We can use matrix multiplication to change the basis from $\mathcal{B}$ to the standard basis, $\mathcal{S}$ and then from $\mathcal{S}$ to $\mathcal{C}$

Change of Basis using the Standard Basis ^theorem

Let $\mathcal{B},\mathcal{C}$ be two ordered, finite, bases of the same vector space, $V$ and let $\mathcal{S}$ be the standard basis. If we have two transition matrices, $P_{\mathcal{S},\mathcal{B}}$ and $P_{\mathcal{S},\mathcal{C}}$ which are the transition matrices for $\mathcal{B}\to \mathcal{S}$ and $\mathcal{C}\to \mathcal{S}$ respectively, then:

$$P_{\mathcal{C},\mathcal{B}}=(P_{\mathcal{S},\mathcal{C}}{)}^{-1}\cdot P_{\mathcal{S},\mathcal{B}}$$

This is useful if you need to change between two very different bases, and want to use the standard basis as an ‘in-between’.