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 >