Say there are 10 houses and we have three pieces of information for each of them, area, nbedrooms, price
I can view this as 10 different vectors in space where there are 3 axes. Basically 10 arrows and each arrow represents area, nbedrooms, price for that particular vector (House). Rather than 3 pieces of information, we can more such information too, d dimensions (axes) and N rows (Houses). So far, I am good
But I am having trouble when I use such visualization in the context of linear regression, where you are predicting price, using those d-1 dimensions and N rows. Especially the column representation equation, Linearly combining areas_column_vector, nbedrooms_column_vector etc to (almost) arrive at price_column_vector
beta1*(areas_column_vector) + beta2*(nbedrooms_column_vector) + ... = prices_column_vector
How should I even visualize these column vectors? Previously, there were N vectors in d axes. Now what? Each column vector has N dimensions now! i.e, house1_area, house2_area, ... houseN_area. Similarly N more dimensions for house1_nbedrooms, house2_nbedrooms,... houseN_nbedrooms. So for each of the (d-1) features used, there would be N dimensions (axes). Because we are taking a linear combination of all these column vectors, they must be in the same multidimensional space. In total there would be N*(d-1) axes! I must be wrong somewhere. Please suggest me a better way of visualizing such column vectors
Even in case of simple linear regression where there is only one feature (say area), beta*area_column_vector = price_column_vector. While visualizing L.H.S of this equation, I see N dimensional (N rows) area_space with 1 dimension for each house. But R.H.S is of a different N dimensional price_space. Where exactly are these two vectors? And what are the axes?