A.5 Vector spaces and lattices
In most undergraduate programs in mathematics the theory of linear algebra and vector spaces is introduced before the theory of groups and rings. This makes sense as vector spaces are easier to comprehend than groups and rings. The reason for putting this section after the preceding sections is simply that we now need fewer axioms to define a vector space.
An n-dimensional vector space over the real numbers can be viewed as the set R^{n} of n-tuples (a_{1}, ¼, a_{n}) of real numbers. The n-tuples are called vectors. There are two basic operations on such a set: addition and scalar multiplication. Addition of vectors is defined as
(a_{1}, ¼, a_{n}) + (b_{1}, ¼, b_{n}) = (a_{1} + b_{1}, ¼, a_{n}+ b_{n}), |
while multiplication of a scalar (real number) r and a vector (a_{1}, ¼a_{n}) is defined as
r ·(a_{1}, ¼, a_{n}) = (r a_{1}, ¼, r a_{n}). |
Given a set S = {v_{1}, ¼, v_{k}} of vectors in R^{n}, the lattice L(S) generated by S is the set of all integer combinations of elements in S. That is, L(S) contains all linear combinations
n_{1} v_{1} + ¼+ n_{k} v_{k}, |
where n_{1}, ¼, n_{k} Î Z. For example, the set of all vectors in R^{3} with integer coordinates is the lattice generated by the unit vectors (1, 0, 0), (0, 1, 0), and (0,0,1).
Formally, a vector space over the field F is a set V of elements called vectors together with an operation + : V ×V® V called addition and an operation ·: F ×V® V called scalar multiplication such that
- V is an abelian group under addition.
- a ·(b ·v) = (ab) ·v for all a,b Î F, v Î V.
- a·(v+w) = a·v + a ·w for all a Î F, v, w Î V.
One may replace F with an arbitrary ring R, but then V is called a module over R and not a vector space. For example, a lattice is a module over Z.
The elements v_{1}, ¼, v_{k} Î V in a vector space are linearly dependent if there are elements a_{1}, ¼, a_{k} Î F not all 0 such that
a_{1} ·v_{1} + ¼+ a_{k} ·v_{k} = 0. |
Otherwise the vectors v_{1}, ¼, v_{k} are linearly independent. The vectors e_{1}, ¼, e_{n} form a basis for V if they are linearly independent and span V, that is, for each v Î V, there are elements a_{1}, ¼, a_{n} Î F such that
a_{1} e_{1} + ¼+ a_{n} e_{n} = v. | (1) |
If there is such a basis, then every basis has the same number of elements; this number is called the dimension of V. In fact, every vector space has a basis, but it need not be finite in general.
With a fixed basis {e_{1}, ¼, e_{n}} of the n-dimensional vector space V, any element v Î V can be written as an n-tuple (a_{1}, ¼, a_{n}), where a_{1}, ¼, a_{n} are the unique elements satisfying (1). Objects in the ordinary three-dimensional space such as planes and lines are easily generalized to arbitrary finite-dimensional vector spaces. For example, an affine hyperplane in V is the set of all points x = (x_{1}, ¼, x_{n}) satisfying the equation
a_{1} x_{1} + ¼+ a_{n} x_{n} = b, |
where a_{1}, ¼, a_{n}, b Î F are some fixed constants (not all a_{i} are zero). The word ``affine'' simply means that the element b does not have to be 0. The concept of a hyperplane generalizes the concept of an affine plane in R^{3}, which has the form
a x + b y + c z = d |
for some constants a,b,c,d Î R. A line in V is a set of the form { av+(1-a)w : a Î F}, where v, w Î V are two different vectors. With w = 0, we obtain a line through the origin.