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Say you are given two vectors x and y represented using different basis sets {ê;} and {f¡}, of the same dimension N. That is, x = y = E yif;. AN. E x;e; and you are told that a given vector x has the representation T (1, 1)" in basis {ê;} and the representation x = {f}}. How might the two basis sets be related? Draw a figure to justify (-1,1)" in basis X = your answer.

Say you are given two vectors x and y represented using different basis sets {ê;} and {f¡}, of the same dimension N. That is, x = y = E yif;. AN. E x;e; and you are told that a given vector x has the representation T (1, 1)" in basis {ê;} and the representation x = {f}}. How might the two basis sets be related? Draw a figure to justify (-1,1)" in basis X = your answer.

Advanced Engineering Mathematics
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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Say you are given two vectors x and y represented using different basis sets {ê;} and {f;}, of the same dimension N. That is, x = y = E y;f;. E xie; and N you are told that a given vector x has the representation x = (1, 1)" in basis {ê;} and the representation x = (-1,1)" in basis {f}}. How might the two basis sets be related? Draw a figure to justify X = your answer.
Transcribed Image Text:Say you are given two vectors x and y represented using different basis sets {ê;} and {f;}, of the same dimension N. That is, x = y = E y;f;. E xie; and N you are told that a given vector x has the representation x = (1, 1)" in basis {ê;} and the representation x = (-1,1)" in basis {f}}. How might the two basis sets be related? Draw a figure to justify X = your answer.
Expert Solution
Step 1

We are given two vectors x and y that are represented using different basis sets e^i and fi^, of the same dimension N.

That is, x=∑i=1Nxiei and y=∑i=1Nyifi.

Since fi^ is the basis, we can write ei  as a linear combination of fi.

That is, ei=∑j=1∞aijfj.

Thus, 

                 x=∑i=1Nxiei⇒x=∑i=1Nxi∑j=1Naijfj⇒x=∑i=1N∑j=1Nxiaijfj

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