Are there any motivation of dual space and annihilator PURELY in linear algebra?

Are there any motivation of dual space and annihilator PURELY in linear algebra? I can prove some relate theorem about it, but I totally can't get what I am doing. It looks like I am, including the textbook writes were, totally just playing some game on the symbols - define dual space without motivation, define annihilator without reasons, and define $T^t=g\mapsto g\circ T$ without any necessity.

I have seen some people say we need that because it is useful in functional analysis, or category theory. However, I think there should be some persuasive reason that why should we define, and what is the motivation of defining them purely in the linear algebra field. Don't tell me "if you grow older, you will know it." If the reasons for it are all in the advanced courses, then why the authors define it and use it in Linear Algebra?

Definitions in my books:

linear functional: a function from a vector space $V$ to a field $F$

dual space on $V$: all linear functionals from $V$ to $F$