Abstraction trades pictures for generality

Textbooks phrase linear algebra axiom-first so theorems apply to every certified vector space . You prove once in the abstract language, then verify axioms in each new domain.

Abstract proofs ideally never mention arrows explicitly. If someone defines exotic "pi creatures" with a custom $+$, they must still verify all vector space axioms before importing the theorems.

Quotient spaces and direct sums illustrate that the vector-space menu extends far beyond $\mathbb{R}^2$. Machine learning practitioners still care: batch tensors live in finite-dimensional spaces, while function spaces appear in kernels, regularization, and operators on models.

When you meet a new object, ask only whether addition and scalar multiplication behave; if yes, import the theorems. If not, linear algebra still helps locally via Jacobians and best linear approximations.

Check your understanding. The tasks below rest on these ideas: Correct: axiomatic proofs use only $+$, scaling, and the axioms, so they transfer to any vector space. Not quite: they still use addition and scalars, and pictures are optional intuition. Correct: the theorems hold once the axioms are checked. Not quite: a determinant is not required, the axioms are not automatic, and the objects need not be arrows. Correct: they are further examples of vector spaces beyond the familiar coordinate ones. Not quite: vector spaces are not limited to $\mathbb{R}^2$, matrices, or finite dimension. Correct: the same vector-space language covers finite-dimensional tensors and infinite-dimensional function spaces alike. Not quite: data is not all 2D, networks rely heavily on linear algebra, and abstraction does not remove computation.