An orthogonal projection matrix in $ \Bbb{R}^{3} $.

Question

Consider the vector space $\mathbb{R^3}$ with usual inner product. Find the orthogonal projection matrix on the xy plane.

I've found sometimes the orthogonal projection of a vector in a given subspace, but in this case I do not know how to proceed.

Answer 1

The first two vectors $e_x$ and $e_y$ are invariant under the projection, and the last one is mapped to 0.

Hence the columns of the matrix are, in order;

$$ \begin{pmatrix} 1\\ 0 \\0 \end{pmatrix} ; \begin{pmatrix} 0\\ 1 \\0 \end{pmatrix} ; \begin{pmatrix} 0\\ 0 \\0 \end{pmatrix} $$

Answer 2

Well this is just asking you to think about what happens. If you project onto the xy plane you will have a $z=0$ component

i.e. you will have your vector multiplied by a matrix of the form

$$ A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} $$

Is this all that you are looking for?