Vector space

A vector space is a collection of objects that can be added together and multiplied by scalars while staying in the same collection. The objects might be arrows, lists of numbers, matrices, functions, or quantum states.

The two core operations are:

$$u+v\in V,cv\in V.$$

A linear combination has the form

$$c_1{v}_1+c_2{v}_2+\cdots +c_n{v}_n.$$

A basis is a set of vectors that lets every vector in the space be written uniquely as a linear combination. The number of basis vectors is the dimension, when the space is finite-dimensional.

Why it matters

Susskind’s quantum mechanics starts by treating states as vectors. A spin state can be a two-component vector; a Wavefunction is more like a vector in an infinite-dimensional function space. Once this is familiar, Hilbert space, operators, and eigenvectors become much less mysterious.

Mini example

In ${\mathbb{R}}^2$, the usual basis vectors are

$$e_1=(1,0),e_2=(0,1).$$

Any vector $(a,b)$ equals $a{e}_1+b{e}_2$.

Common pitfalls

• A vector space is defined by operations, not by visual arrows.
• Basis vectors are choices, not usually physical objects.
• Components change when the basis changes; the vector is the underlying object.

Source trail: Susskind The Theoretical Minimum index; reference book: Quantum Mechanics: The Theoretical Minimum by Leonard Susskind and Art Friedman.