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I'm studying PCA and my professor said something about finding the linear regression by doing the dot product of both axis. Could someone explain to me why? The dot product returns a number. What's the relationship between that number and the linear regression?

In my example, I have two vectors

$stat\_grade = [0,1,3,7,10]$

$physics\_grade = [1,5,8,10,10]$

The first step is normalizing them:

$ \frac{stat\_grade - mean(stat\_grade)}{std(stat\_grade)} = [-1.69131435 -0.52489066 0.34992711 0.93313895 0.93313895]$

$ \frac{physics\_grade - mean(physics\_grade)}{std(physics\_grade)} = [-1.11613741 -0.85039041 -0.3188964 0.7440916 1.54133261]$

Computing the dot product between the normalized vectors will return $4.355129045906131$

I can't understand how a single number can help me to find the linear regression

Thanks for the response!

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  • $\begingroup$ dot product of normalised vectors gives the cosine of the angle one has with the other, thus gives the linear factor in a linear model of one as function of the other $\endgroup$
    – Nikos M.
    Jan 24, 2021 at 17:19
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    $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$
    – Nikos M.
    Jan 24, 2021 at 17:27
  • $\begingroup$ Therefore the dot product gives the slope of the line? $\endgroup$ Jan 24, 2021 at 17:30
  • $\begingroup$ yes, exactly, gave referencve for simple case, to see it better $\endgroup$
    – Nikos M.
    Jan 24, 2021 at 17:31
  • $\begingroup$ I got it but i will need to find the y intercpet to plot the line right? $\endgroup$ Jan 24, 2021 at 17:52

1 Answer 1

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I think the professor might have meant the closed-form solution of linear regression.

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  • $\begingroup$ These are also called normal equations (for reference) $\endgroup$
    – liakoyras
    Nov 18, 2022 at 22:07

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