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.

Tobias Ali 2021-02-19 Answered
To explain^ Why the beta-coordinate vectors of β=b1,...,bn
are the columns e1,...,en
of the n×n identity matrix.
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Expert Answer

Benedict
Answered 2021-02-20 Author has 108 answers

Consider a vector x in V such that, x=c1b1+c2b2+...+cnbn
The coordinates of x relative to basis β=b1,b2,...,bn ,
also called beta-coordinates of x are given by, [x]β=[c1...cn]
Since β=b1,...,bn from a basis for V.
Thus, the vectors b1,...,bn are linearly independent.
Therefore, if any vector bk is to be written in terms of b1,...,bn
That is, an arbitrary vector bk
can be written as bk=0b1,...+1bk+...+0bn.
Here, k varies from 1 to n.
Thus, the beta-coordinates of b1,...,bn
in this case are {[c1...cn]}
The matrix formed by these beta-coordinates is [1001]
That is, beta-coordinate vectors of β=b1,...,bn
are the columns e1,...,en of the
nn identity matrix.

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