This shift from coordinates to vectors is the foundational act of linear algebra. It allows us to think about geometric objects—lines, planes, rotations, stretches—in a coordinate-free way. A vector space $\mathbbR^n$ is the set of all such vectors. But the abstraction goes deeper: a vector space doesn't have to be $\mathbbR^n$. It can be the space of all $2 \times 2$ matrices, the space of all polynomials of degree less than 3, or even the space of all continuous functions on the interval $[0,1]$. These are all vector spaces because they satisfy the same ten axioms: closure under addition and scalar multiplication, the existence of a zero vector, distributivity, and so on.
Sometimes you don't need a 400-page book; you just need a cheat sheet. Stanford University provides a concise review that covers the essential operations used in machine learning. linear algebra pdf
While computers do the heavy lifting, manually performing a This shift from coordinates to vectors is the
Critical for understanding if a system has a unique solution. But the abstraction goes deeper: a vector space
This is an open-source textbook available as a free PDF. It includes extensive exercises and is designed for students who are new to formal proofs. Self-learners on a budget. 4. "Linear Algebra Step by Step" (Cheat Sheets)