This is a short summary of the key terms from the excellent video, “A more formal understanding of functions”, from the Linear algebra section on Khan Academy.

At its core, a function is just a mapping of one set to another set. So, for example, for every element in set X, we associate an element in set Y. And then we call this association a function.

The origin set is called the domain. The destination set is called the co-domain. These sets are part of the function definition, although oftentimes people don’t mention them.

A more concrete example might be

\[f(x) = x^2\]

Alternatively, this may be written as

\[f: x \mapsto x^2\]

The second notation perhaps expresses the association better. This function maps the domain of real numbers to the co-domain of real numbers. Or, in other words, for this function, we can put in any real number and it will give us a real number back.

The subset of the co-domain that is actually mapped is called the range. So, for example, if we had a function g, which takes in any two-dimensional real number and returns 2, the codomain may be the set of real numbers, but the range would be limited to 2.

Vector Transformations

When the co-domain is a one-dimensional real number, the function is called a scalar valued function or a real valued function. The example above is a scalar valued function because $f: x \mapsto x^2$ maps $\mathbb{R} \mapsto \mathbb{R}$.

When the co-domain maps to multi-dimensional real number, the function is referred to as a vector valued function. Or, in other words, when the co-domain is $\mathbb{R}^m$, where $m > 1$, this is a vector valued function. People sometimes refer to this as a vector transformation, but essentially a transformation is simply a function.

Key Terms

  • function
  • domain
  • co-domain
  • range
  • scalar valued function
  • vector valued function
  • transformation