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I can't understand why a finite dimensional vector space is a smooth manifold. I think it's a trivial thing, but I can't deal with it.

Let $V$ a finite-dimensional real vector space. $V$ is a topological manifold of dimension $n$. This because $V\simeq\mathbb{R}^n$.

Now, each basis $(E_1,\dots, E_n)$ for $V$ defines a basis isomorphism $E\colon\mathbb{R}^n\to V$ by $$E(x)=\sum_{i=1}^nx_iE_i.$$

This map is a homeomorphism, then $\big(V,E^{-1}\big)$ is a chart. if $\big(F_1,\dots,F_m\big)$ is any other basis and $F(x)=\sum_jx_jF_j$ is the correspondig isomorphism, then there is some invertible matrix $\big(A_j^i \big)$ such that $E_i=\sum_j A_j^{i} F_j$ for each $i$. Up to this point it seems to have been applied, without further explanation.

The transition map between the two charts $\big(V, E^{-1}\big)$, $\big(V, F^{-1}\big)$ is given by $F^{-1}\circ E(x)=\tilde{x}$, where $\tilde{x}=(\tilde{x_1},\dots, \tilde{x_n})$ is determined by $$\sum_{j=1}^n \tilde{x_j} F_j=\sum_{i=1}^{n} x_iE_i=\sum_{i,j}^{n}x_iA_j^{i} F_j$$

Question. Why this thing?

It follows that $\tilde{x}=\sum_i A_i^{j}x_i$.

Thus, the map sending $x$ to $\tilde{x}$ is an invertible linear map.

Question Why the map sending $x$ to $\tilde{x}$ is an invertible linear map?

Thanks!

Jack J.
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$\tilde{x}=F^{-1}\circ E(x)$, so $F(\tilde{x})=E(x)$, i.e. by definition of $F$ and $E$ $$ \sum_j\tilde{x}_j F_j=\sum_jx_j E_j$$

and since $E_j=\sum_i A^i_j F_j$

$$ \sum_j\tilde{x}_j F_j=\sum_jx_j E_j=\sum_{j,i}x_j A^i_j F_j$$

you might recognize that $\tilde{x}_j=\sum_i A^i_j x_i$ is just the matrix vector multiplication $\tilde{x}=Ax$. Since $A$ maps a basis to a basis, $A$ is an invertible matrix.

user438666
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  • @user438666Thanks for your answer. The last question: the function $E$ is continuous because the its component functions, being constant, are continuos? – Jack J. May 02 '19 at 17:55
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    $E$ is continuous because it's linear and on a finite dimensional space, I don't quite understand what you mean by component function – user438666 May 02 '19 at 18:15