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I would like to clarify if my understanding is correct. In the traditional X-Y coordinate system, our choice of basis vectors are $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$ and when you I have a point $(3, 2)$, I am essentially doing, "a linear transformation"

$$\begin{bmatrix}3 \\2 \end{bmatrix}.\begin{bmatrix}1 & 0\\0 & 1 \end{bmatrix} = (3, 2)$$

Now, after I find say a particular principal component. Every principal component that I find happens to be:

  1. An eigenvector MUCH like $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$
  2. A basis vector MUCH like $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$
  3. Have unit magnitude MUCH like $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$

The only difference is that, since the principal components are a NEW BASIS! In other words, they represent a different coordinate representation (like how we have X-Y coordinate system with $\vec{i}$ and $\vec{j}$)

We have now FOUND a new coordinate systems whose basis vector is an eigenvector (assume you take a single component), and we would like to translate how $(3, 2)$ which WAS represented using $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$, to how it SHOULD BE represented with the new basis vector (eigenvector)

And that is why, to find what $(3, 2)$ is in the new coordinate system represented by their basis/eigenvectors, we perform a dot product between the point we want to transform i.e $(3, 2)$ with the eigenvector.

Is my intuition ACCURATE? Did I correctly explain the meaning of why we do a dot product between principal components and our data points?

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  • $\begingroup$ Since your account doesn't even mention "dot product," it's hard to see how it could explain anything about it. $\endgroup$ – whuber Jan 9 at 23:59
  • $\begingroup$ @whuber Projection of a point onto its eigen vector by doing a dot product which is a linear transformation.. I'm just trying to reason to myself what doing this tries to accomplish $\endgroup$ – imperialgendarme Jan 10 at 0:03

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