$\text{proj} _{u_1}v + \text{proj} _{u_2}v = \text{proj} _{U}v$

Question

Let $U = $span$\{u_1,u_2\}$ , $u_1,u_2$ are orthogonal and $v$ be a vector. Does the following holds true? $$\text{proj} _{u_1}v + \text{proj} _{u_2}v = \text{proj} _{U}v$$

Answer 1

Yes, this is true. It is enough to verify that the equation holds for $v$ of the form $au_1+bu_2$ and the equation holds for this $v$ because $proj_{u_1} (au_1+bu_2)=au_1$ (in view of orthogonality) and $proj_{u_2} (au_1+bu_2)=bu_2$

Answer 2

The answer is no. For instance, take $$ v = u_1 = (1,0), \quad u_2 = (1,1) $$