Vector space
From CGAFaq
A vector space is a kind of algebraic structure, consisting of scalars, vectors, and a few operations satisfying essential properties. Vector spaces occur everywhere in computer graphics.
The scalars of a vector space are most often the real numbers, but more generally may be any field, such as the complex numbers. Loosely speaking, a field is a set of objects that we can add, subtract, multiply, and divide. If we give up division we have a ring (like the integers), and instead of a vector space we will have a module.
The vectors of a vector space also come with operations, and form an additive group. Loosely, a group is a set of objects we can add and subtract among themselves. A typical example would be translations.
We do not assume that we can multiple vectors alone, but we insist that we can multiply vectors by scalars. For example, we can scale a translation to be 50% longer or 50% shorter. Scaling is required to distribute over vector addition.
- a(u+v) = a u + a v
Mappings (functions) that preserve vector algebra (summing and scaling) are called linear. Vector spaces are also sometimes called linear spaces.
A linear combination is a weighted sum of vectors, a1v1 + ⋯ + anvn.
If a finite list of vectors has the property that no vector in the list can be written as a linear combination of those preceeding it, we say the vectors are linearly independent; otherwise, they are linearly dependent.
The span of a given a list of vectors is the set of all vectors that are linear combinations of them.
A list of linearly independent vectors that spans the vector space is a basis (sometimes, ordered basis).
If a vector space has a finite basis, then every basis has the same length, called the dimension of the vector space. Otherwise, the space is infinite dimensional.
Given a basis, each vector in the space is a unique linear combination of the basis vectors, and the list of weights gives the coordinates of that vector with respect to the given basis.
