Linear Algebra

Topic: Orthonormal Matrices 289invariant under some group of transformations. The word ‘group’ here meansmore than just ‘collection’, but that lies outside of our scope.)We can use linear algebra to characterize the distance-preserving maps ofthe plane.We must first observe that there are distance-preserving transformations ofthe plane that are not linear. The obvious example is this translation.(xy)↦→(xy)+(10)=(x + 1yHowever, this example turns out to be the only example, in the sense that if f isdistance-preserving and sends ⃗0 to ⃗v 0 then the map ⃗v ↦→ f(⃗v) −⃗v 0 is linear. Thatwill follow immediately from this statement: a map t that is distance-preservingand sends ⃗0 to itself is linear. To prove this equivalent statement, lett(⃗e 1 ) =(ab)t(⃗e 2 ) =for some a, b, c, d ∈ R. Then to show that t is linear we can show that it can berepresented by a matrix, that is, that t acts in this way for all x, y ∈ R.( ) ( )x t ax + cy⃗v = ↦−→(∗)y bx + dyRecall that if we fix three non-collinear points then we can determine any pointby giving its distance from those three. So we can determine any point ⃗v inthe domain by its distance from ⃗0, ⃗e 1 , and ⃗e 2 . Similarly, we can determineany point t(⃗v) in the codomain by its distance from the three fixed points t(⃗0),t(⃗e 1 ), and t(⃗e 2 ) (these three are not collinear because, as mentioned above,collinearity is invariant and ⃗0, ⃗e 1 , and ⃗e 2 are not collinear). In fact, becauset is distance-preserving, we can say more: for the point ⃗v in the plane that isdetermined by being the distance d 0 from ⃗0, the distance d 1 from ⃗e 1 , and thedistance d 2 from ⃗e 2 , its image t(⃗v) must be the unique point in the codomainthat is determined by being d 0 from t(⃗0), d 1 from t(⃗e 1 ), and d 2 from t(⃗e 2 ).Because of the uniqueness, checking that the action in (∗) works in the d 0 , d 1 ,and d 2 casesdist((xy),⃗0) = dist(t((xy)(cd), t(⃗0)) = dist((we assumed that t maps ⃗0 to itself)( )( )xxdist( ,⃗e 1 ) = dist(t( ), t(⃗e 1 )) = dist(yy)(()ax + cybx + dy))ax + cybx + dy,⃗0)( )a, )band( )( )( ) ( )xxax + cy cdist( ,⃗e 2 ) = dist(t( ), t(⃗e 2 )) = dist(, )yybx + dy d