Definition:

Let F be a nonempty set, with $+,\times:F^2\to F$ two binary operations on F.

We call $(F,+,\times)$ a field if the following conditions are met:

• $(F,+)$ is a commutative group, with identity which we’ll call 0.
• $(F\smallsetminus \{0\},\times)$ is a commutative group with identity 1.
• $a(b+c) = ab+ac$ for all $a,b,c\in F$.

While you probably studied linear algebra by looking at matrices of real numbers, it turns out the “of real numbers” part wasn’t really all that important.

The complex numbers also form a field. If you instead study matrices of complex numbers, you’d find that most of the study is not very different! Almost all the theorems and methods are the same.

And moreover, we could have studied matrices in just about any field. Most of the results would be unaffected.

Therefore we denote by $F^{m\times n}$ the set of all m by n arrays of elements from F, for $m,n\in\Bbb Z^+$.

Addition and multiplication on $F^{n\times n}$ is defined as with regular matrices.

Definition:

Let $n\in\Bbb Z^+$ and define the general linear group of degree n by

$$ GL_n(F) = \{M\in F^{n\times n}: \text{det}(M)\ne 0\} $$

In effect this is just the set of all invertible matrices.