Isometries and the Linear Algebra of Quadratic

0.1. 2D. In the context of linear algebra a plane is a two-dimensional real vector space. A basis for a plane consists of any two vectors E1,E2 which span the plane. ‘Spanning’ means that any vector v⃗ in the plane can be written as v⃗ = uE1 + vE2 with u, v ∈ R. It is a theorem that for two vectors in the plane “spanning” is equivalent to being “linearly independent”. The ‘linearly independent part” implies that this expression for v⃗ is unique. Thus a choice of basis gives us coordinates on the plane. Any linear transformation T of the plane to itself is uniquely determined by where it sends a basis. Suppose that E1 ↦ U⃗1 and E2 ↦ U⃗2 Then uE1 + vE2 ↦ uU1 + vU2, i.e T (uE1 + vE2) = uU1 + vU2. Usually we fixate on a “standard basis” which we are taught to identify with (1,0) and (0,1) or i and j or e1 and e2. If U1 = ae1 + ce2, U2 = be1 + de2 then the matrix of this linear transformation is