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Let $x_1, \dots, x_N$ be the standard coordinate functions on $\mathbb{R}^N$, and let $X$ be a $k$-dimensional submanifold of $\mathbb{R}^N$. Prove that every point $x \in X$ has a neighborhood on which the restrictions of some $k$-coordinate functions $x_{i_1}, \dots, x_{i_k}$ form a local coordinate system.

I have proved the lemma:

Let $e_1, \dots, e_N$ be the standard basis for $\mathbb{R}^N$. The projection of $T_x(X)$ onto the subspace spanned by $e_{i_1}, \dots, e_{i_k}$ is bijective for some choice of $i_1, \dots, i_k$.

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    $\begingroup$ $X$ is supposedly a $k$-dimensional submanifold of $\mathbb{R}^n$? Then $T_x(X)$ is a $k$-dimensional subspace of $\mathbb{R}^n$. A linear map from one $k$-dimensional subspace of a vector space to another $k$-dimensional subspace of that ambient vector space may be bijective or not. $\endgroup$ – Daniel Fischer Jul 2 '13 at 23:54
  • $\begingroup$ Oh, yes, $X$ is a $k$-dimensional submanifold. Thanks, that helps a lot! $\endgroup$ – 1LiterTears Jul 2 '13 at 23:58
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    $\begingroup$ Hint: Think about the $N\times k$ matrix whose columns are a basis of $T_xX$. $\endgroup$ – Ted Shifrin Jul 3 '13 at 0:02
  • $\begingroup$ Hi Professor Shifrin! Thanks! Let me think about it! How is your summer? $\endgroup$ – 1LiterTears Jul 3 '13 at 0:03
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Consider the $k \times N$ matrix $M$ with rows as the vectors in $\mathbb{R}^N$ that are linearly independent, hence span $T_x(X)$. The rank of $M$ is $k$. Then apply Gaussian elimination, resulting a matrix $M$ with the columns $i_1, \dots, i_k$ all 0's but a single 1. Hence the projection of $T_x(X)$ onto the subspace spanned by $e_{i_1}, \dots, e_{i_k}$ is a bijection.

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