sin(x) as a vector: Find the normalised vectors w1* , w2* , w3* , ...

martha

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[FONT=&quot]Hi all![/FONT]

[FONT=&quot]I need to complete an assignment and I am kinda stuck, so I could do with some help![/FONT]
[FONT=&quot]This is the info given at the beginning of the assignment:[/FONT]

[FONT=&quot]Fourier series can be seen as a generalisation of the idea of the scalar product. Consider on the one hand the ordinary vectors
v1 = (1, 1) and v2 = (1,-1), and on the other hand the functions w1 = sin(x), w2 = sin(2x), w3 = sin(3x), ..., which we shall think
of as a kind of “vectors” as well.
[/FONT]
[FONT=&quot] [/FONT]

[FONT=&quot]This is the question I am at now:
[/FONT]

[FONT=&quot]Find the normalised vectors w1* , w2* , w3* , ... (above you may have used Pythagoras’s theorem for this step, but now you will need to formulate the normalisation purely in terms of scalar products, using the usual relation between scalar products and lengths).[/FONT]

[FONT=&quot]For one of the previous questions I indeed used pythagoras to get the normalised vectors for v1 and v2. What I did sofar to answer this question is the following:[/FONT]

[FONT=&quot]w1∙w1=|w1|^2
sin(x)∙sin(x)=sin(x)^2= |w1|^2
|w1|=sqrt(sin(x)^2)= sin⁡(x)
w1*=w1/|w1| =sin⁡(x)/sin⁡(x)=1
[/FONT]


[FONT=&quot]I also got 1 as the answer for w2* and w3*, using the same method. However, the next assignment requires me to check that these vectors are orthogonal so I figured my answers cannot be correct. w1*,w2* and w3* will not be orthogonal if they all equal 1, as the scalar product will not be 0, so I am wondering what I should be doing differently.[/FONT]

[FONT=&quot]Can someone please explain to me what I should be doing instead?[/FONT]
 
Hi martha,

The "vector space" you are dealing with is now a space of all possible continuous functions defined on some interval, and the inner product of two functions in this space is defined as an integral over that interval, rather than just being a simple multiplication.

https://en.wikipedia.org/wiki/Inner_product_space#Hilbert_space

Therefore, I think you need to use this integral definition of the inner product to ensure that your basis functions (sinusoids in this case) are orthonormal.

Hope that helps.
 
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