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Find the change of coordinates matrix from B to the standard basis in R^n.

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

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:

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

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