Cardinality of Generator is not Less than Dimension

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Theorem

Let $V$ be a vector space over a field $F$.

Let $\mathcal B$ be a generator for $V$ containing $m$ elements.


Then:

$\map {\dim_F} V \le m$

where $\map {\dim_F} V$ is the dimension of $V$.


Proof

Let $\mathcal B = \set {x_1, x_2, \ldots, x_m}$ be a generator for $G$.


Let $\set {y_1, y_2, \ldots, y_n}$ be a subset of $G$ such that $n > m$.

As $\mathcal B$ generates $G$, there exist $\alpha_{ij} \in F$ where $1 \le i \le m, 1 \le j \le n$ such that:

$\displaystyle \forall j: 1 \le j \le n: y_j = \sum_{i \mathop = 1}^m \alpha_{ij} x_i$

Now let $\beta_1, \ldots, \beta_n$ be elements of $F$.

Then:

$\displaystyle \sum_{j \mathop = 1}^n \beta_j y_j = \sum_{i \mathop = 1}^m \sum_{j \mathop = 1}^n \paren {\alpha_{ij} \beta_j} x_i$

As $n > m$ the result Homogeneous Linear Equations with More Unknowns than Equations can be applied.

That is, there exist $\beta_1, \ldots, \beta_n \in F$ which are not all zero such that:

$\displaystyle \forall i: 1 \le i \le m: y_j = \sum_{j \mathop = 1}^n \alpha_{ij} \beta_j = 0$

That is, such that:

$\displaystyle \sum_{j \mathop = 1}^n \beta_j y_j = 0$

So $\set {y_1, y_2, \ldots, y_n}$ is linearly dependent.


The result follows by definition of dimension.

$\blacksquare$


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