# To explain^ Why the beta-coordinate vectors of beta = {b_{1}, ... , b_{n}} are the columns e_{1}, ... , e_{n} of the n times n identity matrix.

To explain^ Why the beta-coordinate vectors of $\beta ={b}_{1},...,{b}_{n}$
are the columns ${e}_{1},...,{e}_{n}$
of the $n×n$ identity matrix.
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Benedict

Consider a vector x in V such that, $x={c}_{1}{b}_{1}+{c}_{2}{b}_{2}+...+{c}_{n}{b}_{n}$
The coordinates of x relative to basis $\beta ={b}_{1},{b}_{2},...,{b}_{n}$ ,
also called beta-coordinates of x are given by, $\left[x{\right]}_{\beta }=\left[\begin{array}{c}{c}_{1}\\ ...\\ {c}_{n}\end{array}\right]$
Since $\beta ={b}_{1},...,{b}_{n}$ from a basis for V.
Thus, the vectors ${b}_{1},...,{b}_{n}$ are linearly independent.
Therefore, if any vector ${b}_{k}$ is to be written in terms of ${b}_{1},...,{b}_{n}$
That is, an arbitrary vector ${b}_{k}$
can be written as ${b}_{k}=0\cdot {b}_{1},...+1\cdot {b}_{k}+...+0\cdot {b}_{n}.$
Here, k varies from 1 to n.
Thus, the beta-coordinates of ${b}_{1},...,{b}_{n}$
in this case are $\left\{\left[\begin{array}{c}{c}_{1}\\ ...\\ {c}_{n}\end{array}\right]\right\}$
The matrix formed by these beta-coordinates is $\left[\begin{array}{ccc}1& \cdots & 0\\ \cdots & \cdots & \cdots \\ 0& \cdots & 1\end{array}\right]$
That is, beta-coordinate vectors of $\beta ={b}_{1},...,{b}_{n}$
are the columns ${e}_{1},...,{e}_{n}$ of the
$n\cdot n$ identity matrix.