I'm currently taking an honors course in linear algebra and am trying to get a feel for what the dual space actually is and what the motivation behind creating such a space is. From what I understand, the dual space naturally appears when we're trying to solve linear equations:

$$f_1(x_1,x_2,.....x_n) = a_{11} x_{1} + a_2 x_2 + .... a_{1n} x_n = b_1$$ $$\vdots$$ $$\vdots$$ $$f_n(x_1,x_2,.....x_n)=a_{11} x_{1} + a_2 x_2 + .... a_{1n} x_n = b_n$$

where $f_i$ would essentially take the role of a row vector of a matrix. Thus, $f_i$ is the linear map from a vector in $\mathbb{R}^n$ to a linear equation $\mathbb{R}$.

I also understand that this is particularly helpful for defining linear functions from a vector space to $\mathbb{R}$ when there is no canonical basis for your vector space.

What I don't understand is what the dual space is telling us for vector spaces like $\mathbb{R}[x]$ (with the standard basis). I understand that one dual basis for this vector space are the linear differential operators $D^k = \frac{1}{k!}\frac{d^k}{dx^k}\biggr\rvert_{x=0}$ which obviously again maps this the $\mathbb{R}[x]$ to a linear equation in $\mathbb{R}$, but what is the meaning of this linear equation? Does defining the dual space for this vector space give us any insight into $\mathbb{R}[x]$? Does defining the dual space for this vector space make any previously difficult problems easier?

Any extra insight into the use and general purpose of dual spaces would also be greatly appreciated!