$\text{Change of Coordinates}$

$\text{Two Bases}\text{in Vector Space}{\mathbb{R}}^3$

$u=\left(\begin{array}{lll}{u}_0,& u_1,& u_2\end{array}\right)$

$v=\left(\begin{array}{lll}{v}_0,& v_1,& v_2\end{array}\right)$

Any vector in vector space${\mathbb{R}}^3$can be described unambiguously relative to any given basis of${\mathbb{R}}^3$.

This is done by taking the linear combination of the vectors in the given basis.

The set of scalars used in such a linear combination uniquely describes the vector with respect to the given basis.

Those scalars are called the coordinates (or coordinate vector) of the described vector relative to the given basis.

Let$\beta$be any vector in${\mathbb{R}}^3$.

Then,$\beta$can be writen as the linear combination of vectors in basis$u$:

$\beta =j_0{u}_0+j_1{u}_1+j_2{u}_2$

Furthermore,$\beta$can also be written as the linear combination of vectors in ther other basis$v$:

$\beta =k_0{v}_0+k_1{v}_1+k_2{v}_2$

Let${\left[\beta \right]}_u$denote the$$coordinates (or coordinate vector) of$\beta$relative to$u$.

${\left[\beta \right]}_u=\left(\begin{array}{l}{j}_0\\ j_1\\ j_2\end{array}\right)$

Let${\left[\beta \right]}_v$denote the$$coordinates (or coordinate vector) of$\beta$relative to$v$.

${\left[\beta \right]}_v=\left(\begin{array}{l}{k}_0\\ k_1\\ k_2\end{array}\right)$

$\text{Question}$

How do we change a coordinate vector relative one basis into a coordinate vector relative to the other basis?

$\text{Coordinates of Vectors in Basis}v\text{Relative to Basis}u$

Define vectors in basis$v$relative to basis$u$.

$v_0=x_0{u}_0+x_1{u}_1+x_2{u}_2=\left(\begin{array}{lll}{u}_0,& u_1,& u_2\end{array}\right)\left(\begin{array}{l}{x}_0\\ x_1\\ x_2\end{array}\right)=ux$

$v_1=y_0{u}_0+y_1{u}_1+y_2{u}_2=\left(\begin{array}{lll}{u}_0,& u_1,& u_2\end{array}\right)\left(\begin{array}{l}{y}_0\\ y_1\\ y_2\end{array}\right)=uy$

$v_2=z_0{u}_0+z_1{u}_1+z_2{u}_2=\left(\begin{array}{lll}{u}_0,& u_1,& u_2\end{array}\right)\left(\begin{array}{l}{z}_0\\ z_1\\ z_2\end{array}\right)=uz$

From the linear combination expressions above, the coordinate vectors can be written as follows.

${\left[v_0\right]}_u=x=\left(\begin{array}{l}{x}_0\\ x_1\\ x_2\end{array}\right)$

${\left[v_1\right]}_u=y=\left(\begin{array}{l}{y}_0\\ y_1\\ y_2\end{array}\right)$

${\left[v_2\right]}_u=z=\left(\begin{array}{l}{z}_0\\ z_1\\ z_2\end{array}\right)$

${\left[v\right]}_u=\left(\begin{array}{lll}{\left[v_0\right]}_u,& {\left[v_1\right]}_u,& {\left[v_2\right]}_u\end{array}\right)=\left(\begin{array}{lll}x,& y,& z\end{array}\right)$

$x=u^{-1}{v}_0$

$y=u^{-1}{v}_1$

$z=u^{-1}{v}_2$

$\text{Change of Coordinate Matrix}Q$

$Q=\left(\begin{array}{lll}x,& y,& z\end{array}\right)=u^{-1}\left(\begin{array}{lll}{v}_0,& v_1,& v_2\end{array}\right)=u^{-1}v$

Coordinates, relative to$u$,$$of vectors in basis$v$ can be computed by using the change of coordinate matrix$Q$.

${\left[v_0\right]}_u=Q{\left[v_0\right]}_v=Q\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{lll}x,& y,& z\end{array}\right)\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=x$

${\left[v_1\right]}_u=Q{\left[v_1\right]}_v=Q\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=\left(\begin{array}{lll}x,& y,& z\end{array}\right)\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=y$

${\left[v_2\right]}_u=Q{\left[v_2\right]}_v=Q\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)=\left(\begin{array}{lll}x,& y,& z\end{array}\right)\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)=z$

$\text{Write}\beta \text{as Linear Combination of Basis Vectors in Basis}v$

$\beta =k_0{v}_0+k_1{v}_1+k_2{v}_2$

${\left[\beta \right]}_v=\left(\begin{array}{l}{k}_0\\ k_1\\ k_2\end{array}\right)$

$\text{Change Coordinate Vector}{\left[\beta \right]}_v\text{into Coordinate Vector}{\left[\beta \right]}_u$

$\beta =k_0{v}_0+k_1{v}_1+k_2{v}_2$

${\left[\beta \right]}_v=\left(\begin{array}{l}{k}_0\\ k_1\\ k_2\end{array}\right)$

${\left[\beta \right]}_u=k_0{\left[v_0\right]}_u+k_1{\left[v_1\right]}_u+k_2{\left[v_2\right]}_u=\left(\begin{array}{lll}{\left[v_0\right]}_u,& {\left[v_1\right]}_u,& {\left[v_2\right]}_u\end{array}\right)\left(\begin{array}{l}{k}_0\\ k_1\\ k_2\end{array}\right)=\left(\begin{array}{lll}{\left[v_0\right]}_u,& {\left[v_1\right]}_u,& {\left[v_2\right]}_u\end{array}\right){\left[\beta \right]}_v$

${\left[\beta \right]}_u=\left(\begin{array}{lll}{\left[v_0\right]}_u,& {\left[v_1\right]}_u,& {\left[v_2\right]}_u\end{array}\right){\left[\beta \right]}_v=Q{\left[\beta \right]}_v$

$\boxed{{\left[\beta \right]}_u=Q{\left[\beta \right]}_v}$

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