Firstly I would like to point out that this is the question extended from the original question posed by Abhinav Gupta in the link Understanding linear projection in "The Elements of Statistical Learning".

Here is the summary: Confusion starts at the page 11-12 in the book The Elements Of Statistical Learning, where the authors present linear expression of the form $\hat{y} = X^T\hat{\beta}$ including the constant variable 1 in $X$ and the intercept $\hat{\beta}_0$ in the vector of coefficients $\hat{\beta}$. They went on to claim that the hyperplane includes the origin and thus it is a subspace when the constant is included in $X$.

It wasn't at all obvious to me how it goes through origin until I think outside n-dimension(n being the number of input variables) and add one dimension, therefore (n+1)-dimension and everything became clear; namely, if we can add one more degree of freedom, say $x_0$, to our input space we can show that the hyperplane going through the intercept is the special case where $x_0=1$ and it goes through the origin when $x_0=0$.

(gril's visualization in the link above will give you a clear mental picture.)

Then my question is, well ok I get how it can be thought of as a subspace but I am still missing 'why' answer. Is there a technical advantage if we formulate it in such a way? Like, in terms of computation for example? Or is it just a matter of conceptual ease with which we can deal with usual linear form?



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