Meaning of $\{1,2,4\}^5$ in linear algebra

Question

I encountered the expression $\{1,2,4\}^5$ in the context of linear algebra. Could someone please explain what this means.

Answer 1

It is the fifth Cartesian power of the set.

As you know (now) the Cartesian square of a set is the Cartesian product of the set with itself.   The set of all pairs whose members are elements of the set $\{1,2,4\}$.

$\{1,2,4\}^2 = \{1,2,4\}{\times}\{1,2,4\} = \{(1,1), (1,2), (1,4), (2,1), (2,2), (2,4), (4,1), (4,2), (4,4)\}$

The Cartesian power to the fifth exponent is defined in the same way.   The set of all quintuples whose members are elements of the set $\{1,2,4\}$.   There are $3^5$ quintuples and I'm not going to list all $243$ of them.   You grok the idea.

$\{1,2,4\}^5 = \{1,2,4\}{\times}\{1,2,4\}{\times}\{1,2,4\}{\times}\{1,2,4\}{\times}\{1,2,4\} \\ \qquad\qquad = \{(1,1,1,1,1), (1,1,1,1,2), \ldots, (4,4,4,4,4)\}$