1. 4. Topic Cramer’s Rule Speed of Calculating Determinants Projective Geometry

2. Cramer’s Rule ~ Compare the sizes of these shaded boxes.

3. Cramer’s Rule: If |A|  0, then A x = b has a unique solution give by where B i is obtained by replacing column i of A by b. Example: →

4. Exercises 4.Topic.1. 1. The first picture in this Topic (the one that doesn’t use determinants) shows a unique solution case. Produce a similar picture for the case of infintely many solutions, and the case of no solutions.

5. Speed of Calculating Determinants Computational complexity: P & NP problems. Speed of calculation inversely  Number of arithematic operations required. Permutation expansion: Sum of n! terms each involving n multiplications. Stirling’s formula: ln n!  n ln n  n = ln (n/e) n → n!  n n. Row reduction: 2 nested loops of n  n 2 operations.

6. Projective Geometry Perspective: Map (project) the 3-D scene to a 2-D image. Parallel lines seem to meet at the vanishing point. Central projection: from a single point to the canvas. The study of the effects of central projections is projective geometry. Non-orthogonal, non-isometric.

7. 3 types of central projection: Movie projector: ( Mapper P pushes S out to I ) Painter: (Mapper P pulls S back to I ) Pinhole: (Mapper P in- between S & I )

8. Railroad tracks that appear to converge to a point. Push out Pull backIn between. Green line has no image. Vanishing point not image of point on S.

9. For any nonzero v  R 3, the associated point v in the projective plane RP 2 is the set Projective geometry: Points on the same line passing through 0 belong to the same class. Each kv is a homogeneous coordinate vector for v. Deficient of the dome model: Images of points below equator appear behind mapper P. Remedy: Identify antipodal points on sphere. Dome model: v is chosen to be on the upper hemi-sphere. (This device is used in the study of SO(3) & SU(2) groups.)

10. The intersect of a plane through 0 with the sphere is a great circle, which defines a line in RP 2 in the following manner. A plane through 0 in R 3 is the set P (a,b,c) is associated with a line L in RP 2 : where is a row vector (1-form). A point v & a line L are incident (v is on L) iff L v = 0. Proof: The normal of P (a,b,c) is the column vector L T. By definition, any point on P (a,b,c) satisfies L v = 0. Usually, we just refer to this as line L = ( a, b, c ).

11. Duality principle of projective geometry: Interchanging ‘point’ & ‘line’ in any true statement results in another true one. E.g., 2 distinct points in RP 2 determine a unique line.  2 distinct lines in RP 2 determine a unique point. In contrast, 2 distinct lines in R 3 may or may not intersect. Projective geometry is simpler, more uniform, than Euclidean geometry. The projective plane can be viewed as an extension of the Euclidean plane. Points on equator are extra since they do not project onto the Euclidean plane. Equator = ideal line = line at infinity Parallel lines in R 3 intersect at equator of RP 2.

12. Linear algebra is a natural tool for analytic projective geometry. E.g., 3 points t, u, v are collinear (incident in a single line) iff By duality, 3 lines incident on a point iff the representative row vectors are L.D. The shaded triangles are in perspective from O because their corresponding vertices are collinear. Desargue’s Theorem  Equation of a point v is the equation satisfied by any line incident on it, i.e., v 1 L 1 + v 2 L 2 + v 3 L 3 = 0

13. Lemma: If W, X, Y, Z are four points in the projective plane (no three of which are collinear) then there are homogeneous coordinate vectors w, x, y, and z for the projective points, and a basis B for R 3, satisfying the following: Proof: Since W, X, Y are not on the same projective line, any homogeneous coordinate vectors w 0, x 0, y 0 do not lie on the same plane through 0 in R 3 and so form a spanning set for R 3. Proof is complete by setting B =  w, x, y , where w = a w 0, x = b x 0, y = c y 0.

14. Desargue’s Theorem Let Since T 2 is incident on the projective line OT 1, Similarly,

15. The projective line T 1 U 1 is the image of a plane through 0 in R 3 and satisfies → z = 0 For T 2 U 2, we have →  Similarly These are on same projection line since their sum = 0. QED

16. Every projective theorem has a translation to a Euclidean version, although the Euclidean result is often messier to state and prove. Euclidean pictures can be thought of as figures from projective geometry for a model of very large radius.(Projective plane is ‘locally Euclidean’.) The projective plane is not orientable:

17. Exercises 4.Topic.3. 1. What is the equation of this point? 2. (a) Find the line incident on these points in the projective plane. (b) Find the point incident on both of these projective lines.