# Size of Linearly Independent Subset is at Most Size of Finite Generator/Proof 1

## Contents

## Theorem

Let $R$ be a division ring.

Let $V$ be an $R$-vector space.

Let $F \subseteq V$ be a finite generator of $V$ over $R$.

Let $L \subseteq V$ be linearly independent over $R$.

Then:

- $\size L \le \size F$

## Proof

We first consider the case where $L$ is finite.

Let $S \subseteq \N$ be the set of all $n \in \N$ such that:

where:

- $L \setminus F$ denotes the set difference between $L$ and $F$
- $\card L$ and $\card F$ denote the cardinality of $L$ and $F$ respectively.

We use the Principle of Finite Induction to prove that $S = \N$.

### Basis of the Induction

Let $\card {L \setminus F} \le 0$.

Then from Cardinality of Empty Set:

- $L \setminus F = \O$

By Set Difference with Superset is Empty Set:

- $L \subseteq F$

By Cardinality of Subset of Finite Set:

- $\card L \le \card F$

Hence:

- $0 \in S$

This is the basis for the induction.

### Induction Hypothesis

It is to be shown that if $k \in S$ where $k \ge 1$, then it follows that $k + 1 \in S$.

This is the induction hypothesis:

It is to be demonstrated that it follows that:

- For every finite generator $F$ of $V$, if $\card {L \setminus F} \le k + 1$, then $\card L \le \card F$

### Induction Step

This is the induction step:

Assume the induction hypothesis that $n \in S$.

Let $F$ be a finite generator of $V$ such that:

- $\card {L \setminus F} = n + 1$

Let $v \in L \setminus F$.

Let $L' = L \cap \paren {F \cup \set v}$.

- $L' \subseteq L$

By Subset of Linearly Independent Set, it follows that $L'$ is linearly independent over $R$.

Also by Intersection is Subset:

- $L' \subseteq F \cup \set v$

Therefore, by Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set:

- there exists a basis $B$ of $V$ such that:
- $L' \subseteq B \subseteq F \cup \set v$

Since $v \notin F$ is a linear combination of $F$, it follows that $F \cup \set v$ is linearly dependent over $R$.

Therefore:

- $B \subsetneq F \cup \set v$

By Cardinality of Subset of Finite Set:

- $\card B < \card {F \cup \set v} = \card F + 1$

Hence:

- $\card B \le \card F$

We have that:

\(\displaystyle \card {L \setminus B}\) | \(\le\) | \(\displaystyle \card {L \setminus L'}\) | Relative Complement inverts Subsets and Cardinality of Subset of Finite Set | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \card {L \setminus \paren {F \cup \set v} }\) | Set Difference with Intersection is Difference | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \card {\paren {L \setminus F} \setminus \set v}\) | Set Difference with Union | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle n\) | (?) |

Since $n \in S$:

- $\card L \le \card B \le \card F$

Hence:

- $n + 1 \in S$

and so the induction step has been completed.

- $L \setminus F \subseteq L$

From Subset of Finite Set is Finite:

- $L \setminus F$ is finite.

Therefore, we can apply the fact that $S = \N$ to conclude that:

- $\card L \le \card F$

Let $L$ be infinite.

Then by Set is Infinite iff exist Subsets of all Finite Cardinalities, there exists a finite subset $L' \subseteq L$ such that:

- $\card {L'} = \card F + 1$

By Subset of Linearly Independent Set, it follows that $L'$ is linearly independent over $R$.

It is proven above that this is impossible.

Hence the result.

$\blacksquare$

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.4$: Theorem $4.1$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 27$: Theorem $27.9$