Intereting Posts

Probability for pairing up
How to build a subgroup $H\leq S_4$ having order $8$?
Rig module (?) of measures with scalar multiplication given by Lebesgue integration
The limit of a convergent Gaussian random variable sequence is still a Gaussian random variable
How to calculate the integral $\int_{a}^{b} \frac{\ln x}{\sqrt{x^{2}+1}} \, \mathrm dx $
Overlapping Probability in Minesweeper
Symmetric vs. Positive semidefinite
Does there exist an integer $x$ satisfying the following congruences?
Difference Between Limit Point and Accumulation Point?
estimate population percentage within an interval, given a small sample
What are dirichlet characters?
An integral for $2\pi+e-9$
Uncountable linearly independet family in $K^\mathbb{N}$
Order of element in multiplicative group modulo n, where n is not prime
What should be the intuition when working with compactness?

Where can I find a proof of the following general Steinitz exchange lemma:

Let $B$ be a basis of a vector space $V$, and $L\subset V$ be linearly independent. Then there is an injection $j:L\rightarrow B$ such that $L\cup(B\setminus j(L))$ is a disjoint union and a basis of the vector space $V$.

Or can someone suggest a proof of this result? Probably using Zorn’s lemma.

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- How to prove that the evaluation map is a ring homomorphism?
- Zariski topology irreducible affine curve is same as the cofinite topology

Thank you!

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Let $W$ denote the subspace generated by $L$, $\mathcal B$ denote the set of subsets of $B$ whose images in $V/W$ are linearly independent. There is an obvious partial order on $\mathcal B$, namely the one induced by inclusion. Apply Zorn’s Lemma to obtain a maximal element of $\mathcal B$, denoted by $C$, then the disjoint union $C\cup L$ forms a basis of $V$, just as $B=C\cup(B\setminus C)$ do. So there must be a bijection $j:L\xrightarrow{\sim}B\setminus C$. This $j$ will fulfill the requirement.

Choose (by using Zorn’s Lemma) a subset $B’$ of $B$ maximal with the property that $\langle L\rangle\cap \langle B’\rangle=0$. Then $\langle L\rangle+\langle B’\rangle=V$: if $b\in B-B’$ then $\langle L\rangle\cap \langle B’\cup\{b\}\rangle\ne0$ hence there are $l_i\in L$, and $b_i’\in B’$ such that $\sum\alpha_il_i=\sum_i\beta_ib_i’+\beta b$ with $\beta\ne0$, so $b\in \langle L\rangle+\langle B’\rangle$.

Now use that any two bases of a vector space are equipotent. Let $V’=\langle B’\rangle$. The quotient vector space $V/V’$ has the following bases: $L$ and $B\setminus B’$. Then there is a bijection $f:L\to B\setminus B’$ which composed by the inclusion $i:B\setminus B’\to B$ gives us the desired injection, that is, $j=i\circ f$.

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