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, \([x]_{\beta}=\begin{bmatrix}c_{1} \\...\\c_{n} \end{bmatrix}\)

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 \(\left\{\begin{bmatrix}c_{1} \\...\\c_{n} \end{bmatrix}\right\}\)

The matrix formed by these beta-coordinates is \(\begin{bmatrix}1 & \cdots & 0 \\ \cdots & \cdots & \cdots \\ 0 & \cdots & 1 \end{bmatrix}\)

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.