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How does one prove that a that a linear subspace of $\mathbb{R}^n$ is a manifold?

This question arises from Spivak's Calculus on Manifolds, Chapter 5, problem 5-5:

Prove that a k-dimensional (vector) subspace of $\mathbb{R}^n$ is a k-dimensional manifold.

dneug
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1 Answers1

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Let $A \subset \mathbb R^n$ be your $k$-dimensional linear subspace. You have to prove, that for every point $a \in A$ there is a neighbourhood of $a$ that is homeomorphic to the $k$-dimensional Euclidian space. Note, that $A$, as a $k$-dimensional subspace of the $\mathbb R^n$, is itself homeomorphic to the $k$-dimensional Euclidian space (and open).

So for any point $a \in A$ you can choose the whole subspace $A$ as the neighbourhood of $a$ and you are done.

Thekwasti
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