Orthogonal Projection Matrix

Hedge

After 10 years of teaching middle school, I am going back to grad school in math. I haven't seen Linear Algebra in more than a decade, but my first class is on Generalized Inverses of Matrices (what am I thinking?). I have a general "rememberance" understanding of most of the concepts we're discussing, but I get lost on where to start with proofs and such (teaching at such a low level for so long has really rotted my brain).

Anyway, we were asked to find the orthogonal projection matrix from R^3 to the xy-plane. I understand geometrically what we're being asked to do, but I do not know where to start. If anyone could give me an idea, that would be great.

HallsofIvy

"Projection to the xy-plane" just "loses" the z coordinate: that is the point (x,y,z) (or vector xi+ yj+ zk) is mapped to (x, y, 0) (respectively, the vector xi+ yj).

Here's generally, how to determine what matrix corresponds to a linear transformation: what does the transformation do to each basis vector?

[tex]\left(\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right)\left(\beg in{array}{c}1\\0\\0\end{array}\right)= \left(\begin{array}{c}a\\d\\g\end{array}\right)[/tex]

That is, multiplying matrix by i just gives the first column. What does multiplying a matrix by j and k give?

Now, "projection onto the xy-plane" changes i to i, j to j, and k to 0. What are the columns of the matrix?