"To explain^ Why the beta-coordinate vectors of β=b1,...,bn..." was answered by students like you

Tobias Ali

Answered question

2021-02-19

To explain^ Why the beta-coordinate vectors of

β = b_{1}, . . ., b_{n}

are the columns

e_{1}, . . ., e_{n}

of the

n \times n

identity matrix.

Answer & Explanation

Benedict

Skilled2021-02-20Added 108 answers

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 $β = b_{1}, b_{2}, . . ., b_{n}$ ,
also called beta-coordinates of x are given by, $[x]_{β} = [\begin{matrix} c_{1} \\ . . . \\ c_{n} \end{matrix}]$
Since $β = b_{1}, . . ., b_{n}$ 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 ${[\begin{matrix} c_{1} \\ . . . \\ c_{n} \end{matrix}]}$
The matrix formed by these beta-coordinates is $[\begin{matrix} 1 & \dots & 0 \\ \dots & \dots & \dots \\ 0 & \dots & 1 \end{matrix}]$
That is, beta-coordinate vectors of $β = b_{1}, . . ., b_{n}$
are the columns $e_{1}, . . ., e_{n}$ of the
$n \cdot n$ identity matrix.

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