It can be anything that you choose it to be. It can be the whole real line or any proper subset - including disjoint subsets.
It can be matrices, all of the same dimensions (Linear Algebra is based on them) or a whole host of other alternatives.
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They are not. A vertical line is not a function so all linear equations are not functions. And all functions are not linear equations.
Linear equations are a small minority of functions.
Most functions are not like linear equations.
There are linear functions and there are quadratic functions but I am not aware of a linear quadratic function. It probably comes from the people who worked on the circular square.
An equation is a statement that two things are equal. A function is a rule or process that gives you a value if you give it something in its domain (the set of things on which it is defined) as an argument. Functions on numbers that are defined by a rule can usually be expressed by an equation. A linear function is one that can be defined by a linear equation.