# Is any $m$-dimensional strict vector subspeace (so $m<p$) a rotational image of the Euclidean subspace $\mathbb{R}^m$?

Let $$V$$ be a vector subspace of $$\mathbb{R}^p$$ of dimension $$m$$, where $$p < m$$. I'm checking with you if there alawys exists a rotation $$A \in O(p)$$ so that $$A(V) = \mathbb{R}^p \times \{0\}^{p-m}$$, i.e. $$V$$ is the rotational image of a rotation in $$\mathbb{R}^p$$ .

I think the answer is yes, by the following argument, could you please check this?

Let $$V$$ be the span of the orthonormal basis $$\{v_1,...v_n\} \subset \mathbb{R}^p$$. It's enough to show that there exists $$A \in SO(p)$$ so that $$< Av_i , e_j> = 0 \forall m+1 \leq j \leq p$$. Solving this for $$A$$ is basically solving $$m(p-m)$$ equations. But since $$m(p-m) \leq \frac{p^2}{4} \leq \frac{p(p-1)}{2} = dim(O(p))$$ as a manifold. So we can always solve for $$A \in O(p)$$ so that the above equations hold, or equivalently, there exists an $$A \in O(p)$$ so that $$A(V) = \mathbb{R}^p \times \{0\}^{p-m}$$.

Is the above correct?

• How did you get $\frac{p^2}4$? Feb 14, 2020 at 23:22
• Instead, simply pick an orthonormal basis $v_1,\dots, v_m$ of $V$ and extend it to an orthonormal basis $v_1,\dots, v_p$ of $\Bbb R^p$, then simply map $e_i\mapsto v_i$. Feb 14, 2020 at 23:26
• @Berci thanks!! $m(p-m) = mp - m^2 = p^2/4 - (m - p/2)^2 \leq p^2/4$. Feb 14, 2020 at 23:39
• Hint: Find an orthonormal basis in $V$ and complete it to an orthonormal basis in $R^n$. Feb 15, 2020 at 10:34
• The trouble with the "dimension count" that you are using is that you are imposing linear conditions on a non-linear subset of the space of all matrices (namely, on $O(p)$). If you were to do so in ${\mathbb C}^n$ and your nonlinear subset given by some polynomial equations, then indeed, you can prove existence of a solution this way (with some caveats). But when you work with real numbers, this method fails. For instance, consider the single linear condition $x_1=2$ imposed on the unit sphere, $\{x: x_1^2+...+x_n^2=1\}$. Feb 16, 2020 at 16:48