Suppose $B_j = \sum_{i=1}^{r} a_{ij} A_i, j= 1,2,....,t$. How does showing that $B_i's$ are dependent prove that $r \geq n$? I am reading 'Galois Theory by Emil Artin', and while reading the proof of Theorem $2$ on Pg. No. $5$, I couldn't grasp the following step :

Now, let $B_1,. . ., B_t$ be any system of vectors in $V$ where $t > r$,
then there exist $a_{ij}$ such that $B_j = \sum_{i=1}^{r} a_{ij}A_i , j=1,2,...,t$, since $A_{i}^{'}s$ form a generating system. If we can show that $B_1, . . ., B_t$ are dependent, this will give us $r \geq n$.

Here, $A_1 , . .., A_m$ are a generating system of a vector space V of dimension $n$, $r$ is the maximum number of independent elements in the generating system.
I don't get how the above argument gives us $r \geq n$. I now add the neccessary definitions :

Definition $1$ : The dimension of a vector space $V$ over a field $F$  is the maximum number of independent elements of $V$.
Definition $2$: A system $A_1, . .., A_m$ of elements in $V$ is called a
generating system of $V$ if each element $A$ of $V$ can be expressed linearly in terms of $A_1, . .., A_m$.

I feel this is something very basic, but at this point I can't make out what I am missing.

Suppose we can show that $B_1,\ldots,B_t$ are dependent for all choises of $t>r$ and $B_1,\ldots, B_t$. The claim is that we can conclude $r\ge n$.
Indeed, assume $r<n$. Then by definition 1, we can find $t:=n$ independent vectors $B_1,\ldots, B_t$ - and can show that they are dependent!
