Summary

Change of Basis

Transition Matrix

Definition

Let $V$ be a vector space with two bases $B=\{v_1,v_2,\dots ,v_n\}$ and $B^{\prime }=\{v_1^{\prime },v_2^{\prime },\dots ,v_n^{\prime }\}$. The transition matrix $P_{B^{\prime },B}$ from $B$ to $B^{\prime }$ is the matrix whose columns are the coordinates of the vectors in $B^{\prime }$ with respect to $B$.

To convert from the basis $B$ to the basis $B^{\prime }$, we multiply the transition matrix $P_{B^{\prime },B}$ with the coordinates of the vector in the basis $B$: $[v{]}_{B^{\prime }}=P_{B^{\prime },B}[v{]}_B$and the inverse transformation is given by $[v{]}_B=P_{B^{\prime },B}^{-1}[v{]}_{B^{\prime }}$ ($P_{B^{\prime },B}$ is invertible if and only if $B$ and $B^{\prime }$ are linearly independent).

Theorem

Let $V$ be a vector space with bases $B=\{v_1,v_2,\dots ,v_n\}$. Then, the transition matrix $P_{B_S,B}$ from the $B$ to standard basis $B_S$ is the matrix whose columns are the coordinates of the vectors in $B$ with respect to the standard basis. $P_{B_S,B}=\left[\begin{array}{cccc}[v_1{]}_{B_S}& [v_2{]}_{B_S}& \dots & [v_n{]}_{B_S}\end{array}\right]$

Theorem

Let $V$ be a vector space with bases $B$ and $B^{\prime }$, and $P_{B^{\prime },B}$ be the transition matrix from $B$ to $B^{\prime }$. Then, the $P_{B,B^{\prime }}=P_{B_S,B^{\prime }}\cdot P_{B,B_S}$ where $B_S$ is the standard basis of $V$.

Standard Basis Example

Consider the vector space ${\mathbb{R}}^2$ with the standard basis $B^{\prime }=\{(1,0),(0,1)\}$ and the basis $B=\{(1,1),(1,-1)\}$. The transition matrix $P_{B^{\prime },B}$ is

$$P_{B^{\prime },B}=\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$$

To convert a vector $v=(2,3)$ from the basis $B$ to the basis $B^{\prime }$, we multiply the transition matrix with the coordinates of $v$ in the basis $B$:

$$\left[\begin{array}{c}5\\ -1\end{array}\right]=P_{B^{\prime },B}\left[\begin{array}{c}2\\ 3\end{array}\right]$$

And to convert back from the basis $B^{\prime }$ to the basis $B$, we multiply the inverse of the transition matrix with the coordinates of $v$ in the basis $B^{\prime }$:

$$\left[\begin{array}{c}2\\ 3\end{array}\right]=P_{B^{\prime },B}^{-1}\left[\begin{array}{c}5\\ -1\end{array}\right]$$

Non-Standard basis Example

Consider the vector space ${\mathbb{R}}^2$ with the basis $B^{\prime }=\{(1,0),(1,1)\}$ and the basis $B=\{(1,1),(1,-1)\}$. The transition matrix $P_{B_S,B}$ is $P_{B_S,B^{\prime }}=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ and $P_{B_S,B}$ is $P_{B_S,B}=\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$ The transition matrix $P_{B^{\prime },B}$ is then given by $P_{B^{\prime },B}=P_{B_s,B^{\prime }}^{-1}\cdot P_{B_S,B}=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]\cdot \left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]=\left[\begin{array}{cc}0& 2\\ 1& -1\end{array}\right]$