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

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The discussion highlights the abstract nature of concepts such as dual space, dual basis, dual map, and annihilator in linear algebra. While the individual understands the proofs and can solve related problems, there is a struggle to retain the concepts due to a lack of application in other subjects. Key areas where these linear algebra topics are applicable include differential forms, tensors, quantum mechanics, general relativity, and differential geometry. The conversation emphasizes that the mathematical interest in these concepts increases significantly when applied to real-world scenarios, such as the relationship between tangent vectors and derivations in smooth functions, as well as the complexities involved in bilinear multiplications and understanding tensor types.
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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?
 
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Dual spaces and dual bases are used in differential forms, tensors, QM, GR.
 
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.
 
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