# Idea behind change of basis and how it relates to projecting your points onto principal components

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?

• Since your account doesn't even mention "dot product," it's hard to see how it could explain anything about it. – whuber Jan 9 at 23:59
• @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 – Abhishek Babuji Jan 10 at 0:03