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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?

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  • $\begingroup$ How did you get $\frac{p^2}4$? $\endgroup$
    – Berci
    Feb 14, 2020 at 23:22
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    $\begingroup$ 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 $e_i\mapsto v_i$. $\endgroup$
    – Berci
    Feb 14, 2020 at 23:26
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    $\begingroup$ @Berci thanks!! $m(p-m) = mp - m^2 = p^2/4 - (m - p/2)^2 \leq p^2/4$. $\endgroup$ Feb 14, 2020 at 23:39
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    $\begingroup$ Hint: Find an orthonormal basis in $V$ and complete it to an orthonormal basis in $R^n$. $\endgroup$ Feb 15, 2020 at 10:34
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    $\begingroup$ 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, $\{x: x_1^2+...+x_n^2=1\}$. $\endgroup$ Feb 16, 2020 at 16:48

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