Let \(\mathcal {N}\) and \(\mathcal {M}\) be two closed linear subspaces in a complex Hilbert space \(\mathcal {H}\) , and let \(P_{\mathcal {N}}\) and \(P_{\mathcal {M}}\) denote the orthogonal projections onto \(\mathcal {N}\) and \(\mathcal {M}\) , respectively. In this paper, we study linear combinations of \(P_{\mathcal {N}}\) and \(P_{\mathcal {M}}\) . Specifically, we show that if \(P_{\mathcal {N}}\) and \(P_{\mathcal {M}}\) are non-zero and \(\alpha , \beta \in \mathbb {R}\) with \(\alpha \beta \ge 0\) , then \(\begin{aligned} \left\Vert \alpha P_{\mathcal {N}} + \beta P_{\mathcal {M}}\right\Vert = \frac{|\alpha | + |\beta | + \sqrt{(\alpha - \beta )^2 + 4\alpha \beta \left\Vert P_{\mathcal {N}}P_{\mathcal {M}}\right\Vert ^2}}{2}. \end{aligned}\) This provides an affirmative answer to a question recently posed by the second author regarding the norm of linear combinations of orthogonal projections. By setting \(\alpha = \beta = 1\) , we recover the well-known Duncan-Taylor formula for the sum of two projections. We also consider some cases where the scalars are complex.
