Compute the standard matrix representation of the orthogonal projection of R^4 onto the subspace spanned by {(1,1,1,1),(1,2,3,4)}.
Step 1. Find an orthonormal basis for the subspace, e.g. $\displaystyle \textstyle\{\frac12(1,1,1,1),\,\frac1{\sqrt{20}}(-3,-1,1,3)\}.$
Step 2. In R^n, the projection onto the one-dimensional subspace spanned by a single unit vector $\displaystyle (a_1,a_2,\ldots,a_n)$ has $\displaystyle a_ia_j$ as its (i,j)-entry. So for example the projection onto the subspace spanned by $\displaystyle \textstyle\frac1{\sqrt{20}}(-3,-1,1,3)$ is $\displaystyle \frac1{20}\begin{bmatrix}9&3&-3&-9\\ 3&1&-1&-3\\ -3&-1&1&3\\ -9&-3&3&9\end{bmatrix}$. Now do the same thing for the projection onto the subspace spanned by ½(1,1,1,1).
Step 3. If you have an orthonormal set of vectors then the projection onto the subspace spanned by them is the sum of the projections onto the one-dimensional subspaces spanned by the basis vectors. So the answer to the question is the sum of the two matrices from the previous paragraph.