Linear Algebra

Topic: Orthonormal Matrices 293(b) Prove that congruence is an equivalence relation between plane figures.4 In practice the matrix for the distance-preserving linear transformation and thetranslation are often combined into one. Check that these two computations yieldthe same first two components.( ( ) ( a c x e+b d)y f)⎛a c⎞ ⎛ ⎞e x⎝b d f⎠⎝y⎠0 0 1 1(These are homogeneous coordinates; see the Topic on Projective Geometry).5 (a) Verify that the properties described in the second paragraph of this Topic asinvariant under distance-preserving maps are indeed so.(b) Give two more properties that are of interest in Euclidean geometry fromyour experience in studying that subject that are also invariant under distancepreservingmaps.(c) Give a property that is not of interest in Euclidean geometry and is notinvariant under distance-preserving maps.