The exercise [1, p. 34 exercise 2.4] asks to show that if \mathbf{F} is a full rank m \times n matrix with m > n, \mathbf{x} is a m \times 1 vector, and \mathbf{y} is an m \times 1 vector, that the effect of the linear transformation
\mathbf{y}=\mathbf{F}\left(\mathbf{F}^H\mathbf{F}\right)^{-1}\mathbf{F}^H\mathbf{x} (1)

is to project \mathbf{x} onto the subspace spanned by the columns of \mathbf{F}. Specifically, if \{\mathbf{f_1},\mathbf{f_2},...,\mathbf{f_n}\} are the columns of \mathbf{F}, the exercise [1, p. 34 exercise 2.4] asks to show that
(\mathbf{x-y})^{H}\mathbf{f_i}=0, \textrm{ i=1,2,...,n.} (2)

Furthermore it is asked why the transform \mathbf{F}\left(\mathbf{F}^H\mathbf{F}\right)^{-1}\mathbf{F}^H must be idempotent. read the conclusion >