Vectors in n dimensions
Arrows in the plane add tip to tail. What survives when a vector is a list of six numbers, or sixty, with no picture to draw?
Where you are 8 inside
What this area is about
A vector in \mathbb{R}^n is a list of n numbers, added and scaled entry by entry. From those two moves come the central questions of linear algebra: which vectors can be built from a given few, whether any of the few is redundant, and how many are needed.
How these fit together
Operations. Adding and scaling vectors sets up the arithmetic; the dot product adds length, angle and perpendicularity.
Definitions. A linear combination is a sum of scaled vectors. The span of a set is every combination of it. Linear independence means no vector in the set is a combination of the others. Subspaces are sets closed under adding and scaling, such as a line or plane through the origin.
Bases. Basis and dimension defines a basis as an independent spanning list and proves the central result: every basis of a subspace has the same number of vectors. Coordinates and change of basis names a vector by its weights in a basis and converts between bases with an invertible matrix.
A path through it
Adding and scaling first. The dot product can follow at any point after that; nothing else here uses it. Then linear combinations, then span and linear independence; subspaces need span. Basis and dimension need both independence and subspaces, and coordinates come last, also needing a matrix inverse found by row reduction.
This area is the input to matrices as transformations, orthogonality and least squares and eigenvalues.