If \(E_1\) and \(E_2\) are projections onto independent subspaces, then \(E_1 + E_2\) is a projection. True or false?
Solution
The assertion is \(\boxed{\text{false}}\).
1. Construct projections with independent ranges.
On \(\mathbb R^2\), define \[ E_1(x,y)=(x,0), \qquad E_2(x,y)=(0,y-x). \] Their matrices in the standard basis are \[ [E_1]=\begin{pmatrix}1&0\\0&0\end{pmatrix}, \qquad [E_2]=\begin{pmatrix}0&0\\-1&1\end{pmatrix}. \] Direct multiplication gives \[ E_1^2=E_1, \qquad E_2^2=E_2, \] so both are projections [@HK71]. Their ranges are \[ \operatorname{range}E_1=\operatorname{span}(1,0), \qquad \operatorname{range}E_2=\operatorname{span}(0,1), \] which are independent subspaces.
2. Their sum is not a projection.
We have \[ [E_1+E_2] =\begin{pmatrix}1&0\\-1&1\end{pmatrix}. \] But \[ [E_1+E_2]^2 =\begin{pmatrix}1&0\\-2&1\end{pmatrix} \ne \begin{pmatrix}1&0\\-1&1\end{pmatrix}. \] Thus \(E_1+E_2\) is not idempotent and hence is not a projection. Therefore independence of the ranges alone does not make the sum of two projections a projection.