Find the coordinate matrix of x in Rn relative to the standard basis.x = (−6, 12, −4, 9, −8)

Answered: Find the coordinate matrix of x in Rn…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find the coordinate matrix of x in Rⁿ relative to the standard basis.
x = (−6, 12, −4, 9, −8)

Expert Solution

Step 1: Determine the coordinate matrix:

The given vector is x = (−6, 12, −4, 9, −8).

One has to find the coordinate matrix of x relative to the standard basis in $straight real numbers to the power of 5$ .

Step 2: Find the coordinate matrix:

The standard basis of vector space $straight real numbers to the power of 5$ is,

$B equals open curly brackets e subscript 1 equals open parentheses 1 comma space 0 comma space 0 comma space 0 comma space 0 close parentheses comma space e subscript 2 equals open parentheses 0 comma space 1 comma space 0 comma space 0 comma space 0 close parentheses comma e subscript 3 equals space open parentheses 0 comma space 0 comma space 1 comma space 0 comma space 0 close parentheses comma e subscript 4 equals space open parentheses 0 comma space 0 comma space 0 comma space 1 comma space 0 close parentheses comma e subscript 5 equals space open parentheses 0 comma space 0 comma space 0 comma space 0 comma space 1 close parentheses close curly brackets$

The vector x can be written as,

$x equals negative 6 e subscript 1 plus 12 e subscript 2 minus 4 e subscript 3 plus 9 e subscript 4 minus 8 e subscript 5$

Thus, the coordinate matrix relative to standard basis B is,

$open square brackets x close square brackets subscript B equals open square brackets table row cell negative 6 end cell row 12 row cell negative 4 end cell row 9 row cell negative 8 end cell end table close square brackets$

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