* regression
consider the generic problem of fitting a dataset to a linear polynomial

given discrete f: x \rightarrow y

interpolation: y = a + bx

[[1 x_0]            [[y_0]
 [1 x_1]  \cdot [[a]  =   [y_1]
 [1 x_n]]   [b]]     [y_n]]

consider p \in col(A)

then y = p + q for some q \cdot p = 0

then we can generate n \in col(A) by $Az$ and n must be orthogonal to q as well

(Az)^T \cdot q = 0 = (Az)^T (y - p)

0 = (z^T A^T)(y - Ax)
  = z^T (A^T y - A^T A x)
  = A^T Ax
  = A^T y


A^T A = [[n+1      \Sigma_{n=0}^n x_n]
        [\Sigma_{n=0}^n x_n  \Sigma_{n=0}^n x_n^2]]
  
A^T y = [[\Sigma_{n=0}^n y_n]
       [\Sigma_{n=0}^n x_n y_n]]

a_11 = n+1
a_12 = \Sigma_{n=0}^n x_n
a_21 = a_12
a_22 = \Sigma_{n=0}^n x_n^2
b_1 = \Sigma_{n=0}^n y_n
b_2 = \Sigma_{n=0}^n x_n y_n

then apply this with:

log(e(h)) \leq log(C) + rlog(h)
 
* homework 3:

two columns \Rightarrow coefficients for linear regression