Vector spaces and bases

Work over the real numbers first. A vector space has an addition operation and multiplication by real scalars, obeying the familiar rules for associativity, distributivity, a zero vector, and additive inverses. A subspace of an existing vector space contains zero and is closed under addition and scalar multiplication. The line $y=2x$ passes these tests. The parallel line $y=2x+1$ fails immediately because it excludes the origin.

The objects need not be arrows. Polynomials of degree at most two form a three-dimensional real vector space: adding two such polynomials or scaling one never introduces a cubic term. Polynomials of degree exactly two do not form a space, since subtracting a polynomial from itself produces zero. Asking what happens under cancellation is often faster than trying to memorize a catalogue of examples.

A collection spans a space if it can express every vector in that space. It is linearly independent if the only combination producing zero has every coefficient equal to zero. A basis satisfies both conditions. Spanning guarantees that coordinates exist; independence guarantees that they are unique. Three arrows in a plane can span the plane, but their redundant descriptions cannot provide a unique three-coordinate address.

The number of vectors in a basis is the dimension of a finite-dimensional space. The standard basis of the plane is convenient, not mandatory. Choosing two tilted, nonparallel arrows changes the coordinate numbers without changing the underlying displacement. When checking independence, keep the ambient space explicit: two independent vectors form a basis of their span, but they cannot span all of three-dimensional space.

A vector space alone does not specify angles or lengths. An inner product supplies that additional structure. For ordinary real coordinate vectors, the dot product produces lengths and tests orthogonality. An orthonormal basis consists of mutually perpendicular unit vectors; it makes each coordinate a simple dot product. A general basis is still valid, but coordinates then require solving simultaneous equations rather than merely reading projections.

This distinction prevents two later confusions. Quantum amplitudes use complex inner-product spaces, so conjugation enters the geometry. Machine-learning features can be expressed in any basis, but distances and regularization depend on the metric and scaling selected. We are establishing finite-dimensional language here; completeness, infinite-dimensional function spaces, and the quantum measurement rule require additional lessons rather than following automatically from an arrow drawing.