Orthonormality of the yielded basis of the direct sum of two subspaces

Question

if V is the direct sum of two subspaces then is true that by adjoining any two orthonormal basis of of our resp. subspaces yields an orthonormal basis of V? it should be clear from the context i'm only interested in the orthonormality of the yielded basis

Answer 1

No, it is not true. Suppose that you are working on $\mathbb{R}^2$ endowed with its usual inner product. Take $U=\mathbb{R}\times\{0\}$ and $W=\{(x,x)\,|\,x\in\mathbb{R}\}$. Then $\{(1,0)\}$ is an orthonormal basis of $U$, $\left\{\frac1{\sqrt2}(1,1)\right\}$ is an orthonormal basis of $W$, and $\left\{(1,0),\frac1{\sqrt2}(1,1)\right\}$ is not an orthonormal basis of $\mathbb{R}^2$.