Show that if v e ]R is a vector such that Xv = 0, then / + c.v is also a minimizer of the least squares problem, for any c E R. (b) If the columns of X are linearly independent, then what vectors v E...

Show that if v e ]R is a vector such that Xv = 0, then / + c.v is also a minimizer of the least squares problem, for any c E R. (b) If the columns of X are linearly independent, then what vectors v E RP satisfy Xv = 0? (c) If the columns of X are linearly independent, then what vector 4 is the minimizer of the least squares problem. (d) Suppose that p > n. Show that there exist a vector v # 0 such that Xv = 0. Argue based on past b, that there are infinitely many linear regression estimates. Furthermore argue that there is a variable i E { 1, ... , p} such that the regression coefficient of variable i can have different signs, depending on which estimate we choose. Comment on this.