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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;. 2; x;e; and

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;. 2; x;e; and

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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Given only the information provided, can you compare these vectors in any way? Justify your answer with an example. Choose a 2-dimensional vector and use a figure to provide a convincing justification.

Say you are given two vectors x and y represented using different basis sets {êi} and {f;}, of the same dimension N. That is, x = y = E" yifi. E x;e; and
Transcribed Image Text:Say you are given two vectors x and y represented using different basis sets {êi} and {f;}, of the same dimension N. That is, x = y = E" yifi. E x;e; and
Expert Solution
Step 1

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

That is, x=∑i=1Nxiei and y=∑i=1Nyifi.

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

That is, ei=∑j=1Naijfj.

Thus, 

          x=∑i=1Nxiei   =∑i=1Nxi∑j=1Naiifi   =∑i=1N ∑j=1Nxiaiifi

So, we can write the vector x as the linear combination with respect to the basis f^i and the same can be done for y.

Similarly, we can show that both x and y can be represented as the linear combination of with respect to the basis e^i.

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