If P is a projection onto a plane in R3 what is det P Soluti

If P is a projection onto a plane in R3, what is det P?

Solution

Let P:VW be a projection of a finite-dimensional vector space onto a subspace W. If P is not the identity, then dimW < dimV, so the dimV columns of a matrix that represents P, which are the images of a basis for V, must be linearly dependent. Therefore, the determinant of the matrix is 0.