Can someone please prove this showing the steps and explaini

Can someone please prove this? showing the steps and explaining.Thank you!

Solution

Since, V = null ( P ) range ( P ) and

every vector in null ( P ) is orthogonal to every vector in range ( P ) , null ( P ) = ( range ( P ) ).

Now let v V , and so v = u + n, with u range ( P ) and n null ( P ).

Also, v = P v + ( v P v ), with P v range ( P ) and ( v P v ) null ( P ).

Because V = null ( P ) range ( P ), the representation of v as a sum of a vector in range ( P ) with a vector in null(P) is unique, and therefore, P v = u.

Hen ce there exists a subspace u of v such that P v = u.