In my lecture notes I encountered the following result,
For any set $S$ the set $F$ of bijective functions $f:S\to S$ is a group under composition, but is not in general abelian.
Then it was mentioned that:
The set of invertible $n\times n$ matrices over field $\mathbb F$ is a group under matrix multiplication: it is a special case of the above result with $S=\mathbb F^n$ and is non-abelian if $n>1$. It is called the general linear group, $\text{GL}(n,\mathbb F)$.
I am having trouble seeing the connection between the two. How can you think of a matrix as a bijective function when it doesn't have an input? Am I missing something?
Thanks so much in advance 🙂
Best Answer
One important viewpoint is that the whole reason matrices exist is to represent linear transformations. When you think of an $m \times n$ matrix $A$ with entries in a field $F$, you should usually think about the mapping $L_A:F^n \to F^m$ defined by $L_A(x) = Ax$.