Proving Projection Matrices Using Definition | Exam Practice Problems

[renolovexoxo]

I'm doing practice problems for my exam, but I don't really know how to get this one. I'd like to just be able to understand it before my test if anyone can help explain it!

Prove from the definition of projection (given below) that if projv=u(sub A) then A=A^2 and A=A^T. (Hint: for the latter, show that Ax dot y=x dot Ay for all x,y. It may be helpful to write x and y as the sum of vectors in V and V perp.

Def: Let V in Rm be a subspace, and let b be an element of Rm. We define the projection of b onto V to be the unique vector p that is an element of V with the property that b-p is an element of V-perp. We write p=projv b

 

[chiro]

I'm doing practice problems for my exam, but I don't really know how to get this one. I'd like to just be able to understand it before my test if anyone can help explain it!

Prove from the definition of projection (given below) that if projv=u(sub A) then A=A^2 and A=A^T. (Hint: for the latter, show that Ax dot y=x dot Ay for all x,y. It may be helpful to write x and y as the sum of vectors in V and V perp.

Def: Let V in Rm be a subspace, and let b be an element of Rm. We define the projection of b onto V to be the unique vector p that is an element of V with the property that b-p is an element of V-perp. We write p=projv b

Hey renolovexoxo and welcome to the forums.

For the A^2 = A proof, we are given that the projection must be a unique vector which means that if re-project an existing vector that is projected onto some projection subspace that it will be the same. What does this imply about reprojecting an existing projection and how that relates to A^2 (Hint: if Proj(X) = AX, and Proj(AX) = Proj(X) then what is the implications?) You will have to probably use the definitions you have been given for an actual projection to get a proper proof expected by your professor/teacher, but the above idea should give you a better hint.

For the A^T I think the hint is a very good one. I would recommend you expand out the definitions of Ax dot y and x dot Ay where A and Aperp form your initial space: In other words, dim(A) + dim(A_perp) = dim(Rm) = m.