I noticed that the definition of the dimension of a finite-dimensional vector space relies on the fact:
If a vector space V has a finite basis, then every basis of V is finite.
Can someone tell me how to prove it? Thanks.
[Math] How to prove that every basis of a finite-dimensional vector space is finite
linear algebravector-spaces
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Best Answer
Hint Let $B=\{ b_1,.., b_n \}$ be a fixed finite basis.
We can prove something stronger:
Lemma If a set has at least $n+1$ elements, it is linearly dependent.
Proof: Let $v_1,..,v_{n+1} \in S$. Write $$v_i=\sum_{j=1}^n c_{ij}b_j$$
Then, replacing each $v_i$ by the above expression,
$$x_1v_1+...+x_{n+1}v_{n+1} =0$$ leads to an homogeneous system of $n$ equations with $n+1$ unknowns (the coefficient matrix being exactly $C^T$). This homogeneous system has more unknowns that variables, and hence as non-trivial solution.
This proves the linear dependence.
Corollary Any other basis has exactly $n$ elements.
Proof: If $B'$ is any other basis, by Lemma $B'$ has at most $n$ elements. Interchanging $B$ and $B'$ you get also that $B'$ has at least $n$ elements.