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We shall investigate the groups which are associated with the usual (Pythagorean) metric on the vector space Rn. These are the groups which preserve this distance.
It turns out that they involve linear algebra. The reason for this is that straight lines in this geometry can be defined as the shortest paths between points in the metric and since the metric is preserved by the transformations they must then map straight lines to straight lines and we will see (eventually) that this means they involve linear maps (but rather special ones).
Definitions
A map f from Rm to Rn is called linear if it maps a linear combination of vectors to the same linear combination of the images.
That is, if u, v ∈ Rm and λ, μ ∈ R then f(λu + μv) = λf(u) + μf(v) ∈ Rn.
By fixing a basis {b1 , b2 , ... , bn } of the vector space Rn (for example, {(1, 0, 0, ... , 0), (0, 1, 0, ... , ), ... , (0, 0, ... , 0, 1)} ) we can describe the effect of such a map by its matrix. Mf = (aij) where f maps the ith basis element bi to ai1b1 + ai2b2 + ... + ainbn.
Such a transformation is a bijection if it has an inverse map f-1 or equivalently if the determinant of its matrix is non-zero.
The set of all such invertible linear transformations from the vector space Rn to itself is called the General Linear group and is denoted by GL(n, R) or GLn(R) or GL(Rn)
Note that the determinant of a matrix satisfies det(AB) = det(A) × det(B) and so is a group homomorphism from the group GL(n, R) to the group R - {0} under real multiplication. The set of all invertible transformations (or equivalently of invertible matrices) with determinant 1 is then a subgroup of GL(n, R) called the Special Linear group and denoted by SL(n, R).
Remarks
For example, GL(1, R) is just R - {0} and so has dimension 1. SL(1, R) is the set {1, -1} and so has dimension 0.
GL(2, R) = {(a, b, c, d) ∈ R4 | ad - bc ≠ 0 } and is the set of points which "miss" the hypersurface with equation ad = bc and so is an open set in R4. The subgroup SL(2, R) = {(a, b, c, d) ∈ R4 | ac - bd = 1 } and is a 3-dimensional subset of R4.
Definitions
We will denote the norm or length of a vector x by ‖x‖ This is d(x, 0).
A distance preserving linear transformation T is said to be orthogonal. That is ‖T(x)‖ = ‖x‖ for all vectors x.
Such transformations form the orthogonal group O(n).
Remarks
Proof
Since T is length preserving it can't map a non-zero vector to 0 and so the null-space has dimension 0 and T is one-one. It is a standard result about linear transformations from n dimensional spaces to n dimensional spaces that dim(null space) + dim(range) = n and so the dimension of the range of T is n and it is hence an onto map. Thus it is a bijection.
Definition
The Special Orthogonal group SO(n) is the subgroup of O(n) of elements whose matrix has determinant 1.
Remarks
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