Dot Product as Linear Transformation and Duality of Vector-Linear Transformation

So the physical interpretation of a dot product is related to the projection of one vector over the other and multiplying by the length of the other vector.

But how we practically compute dot product is element wise multiplication and summation. So the question is Why does this process of mutiplying the coordinates value and adding them together has to do anything with projection? Ans:

Duality: Linear Transformation and Vector

The dual of a vector is the linear transformation is encoded. <=> The dual of a "linear transformation from some space to one dimension" is a certain vector in the space