\[ \boldsymbol{ B \ = \ \left\{ \Bigg [ \begin{array}{c} 1 \\ -2 \\ 5 \end{array} \Bigg ] , \Bigg [ \begin{array}{c} 3 \\ 0 \\ -1 \end{array} \Bigg ] , \Bigg [ \begin{array}{c} 8 \\ -2 \\ 7 \end{array} \Bigg ] \right\} } \]

The aim of this question is to find the **change-of-coordinates matrix** given a set of** basis vectors**.

A **change-of-coordinates matrix** is such a matrix that mathematically represents the **conversion of basis vectors** from one **coordinate system** to another. A change-of-coordinates matrix is also called a **transition matrix**.

To perform this conversion, we **simply multiply the given basis vectors** one by one **with the transition matrix,** which gives us the basis vectors of the new coordinate system.

If we are **given a set of $ n $ basis vectors**:

\[ \left\{ < v_1 > , \ < v_2 > , \ … \ , \ < v_n > \right\} \]

Now if we have to convert them to a standard $ R^n $ coordinates, the **change-of-coordinates matrix** is simply given by:

\[ \left[ \begin{array}{ c c c c } | & | & & | \\ v_1 & v_2 & … & v_n \\ | & | & & | \end{array} \right] \]

## Expert Answer

**Given:**

\[ B \ = \ \left\{ \Bigg [ \begin{array}{c} 1 \\ -2 \\ 5 \end{array} \Bigg ] , \Bigg [ \begin{array}{c} 3 \\ 0 \\ -1 \end{array} \Bigg ] , \Bigg [ \begin{array}{c} 8 \\ -2 \\ 7 \end{array} \Bigg ] \right\} \]

**Here:**

\[ v_1 \ = \ \Bigg [ \begin{array}{c} 1 \\ -2 \\ 5 \end{array} \Bigg ] \]

\[ v_2 \ = \ \Bigg [ \begin{array}{c} 3 \\ 0 \\ -1 \end{array} \Bigg ] \]

\[ v_3 \ = \ \Bigg [ \begin{array}{c} 8 \\ -2 \\ 7 \end{array} \Bigg ] \]

The **transition matrix** $M$ in this case can be found using the **following formula**:

\[ M \ = \ \left[ \begin{array}{ c c c } | & | & | \\ v_1 & v_2 & v_3 \\ | & | & | \end{array} \right] \]

**Substituting values:**

\[ M \ = \ \left[ \begin{array}{ c c c } 1 & 3 & 8 \\ -2 & 0 & -2 \\ 5 & -1 & 7 \end{array} \right] \]

## Numerical Result

\[ M \ = \ \left[ \begin{array}{ c c c } 1 & 3 & 8 \\ -2 & 0 & -2 \\ 5 & -1 & 7 \end{array} \right] \]

## Example

Calculate the **standard change of coordinates matrix** for the following basis vectors:

\[ \boldsymbol{ B \ = \ \left\{ \Bigg [ \begin{array}{c} a \\ b \\ c \end{array} \Bigg ] , \Bigg [ \begin{array}{c} d \\ e \\ f \end{array} \Bigg ] , \Bigg [ \begin{array}{c} g \\ h \\ i \end{array} \Bigg ] \right\} } \]

The required **transition matrix** is given by:

\[ M \ = \ \left[ \begin{array}{ c c c } a & d & g \\ b & e & h \\ c & f & i \end{array} \right] \]