Where in STEM can I expect to use dual basis, dual map, annihilator?

[zenterix]

I find the topics of dual space, dual basis, dual map, and annihilator quite abstract.

I understand the proofs in the book I am reading (Linear Algebra Done Right), and I can solve problems. But after a few weeks without doing any linear algebra I forget what these concepts are and the reason is that I haven't seen them in other subjects in the past and I don't see them in any other subject that I am currently studying (thermodynamics, special relativity, differential equations, electromagnetism).

Where in STEM can I expect to see these linear algebra topics used?

 

[Hill]

Dual spaces and dual bases are used in differential forms, tensors, QM, GR.

 

[fresh_42]

In general: differential geometry, tensor algebra, i.e. everywhere in physics. You are right, the concept is mathematically a bit boring if you only consider finite-dimensional vector spaces with an inner product. This changes if you look at the correspondence of tangent vectors and derivations on ##C^\infty (\mathbb{R}^n)##,
$$
v\longleftrightarrow \left(v(f)=\left. \dfrac{d}{dt}\right|_{t=0}f(p+tv)\right)
$$
or investigate the computational complexity of bilinear multiplications,
$$
(x,y) \longmapsto \operatorname{min}\left\{r\, \left| \,x\cdot y =\sum_{\rho=1}^r u_\rho(x)v_\rho(y)W_\rho \right.\right\}
$$
or simply try to understand what a ##(2,1)##-tensor is.