1. 1. Vector Space 24. February 2004

2. Real Numbers R. Let us review the structure of the set of real numbers (real line) R. In particular, consider addition + and multiplication £. ( R,+) forms an abelian group. ( R, £ ) does not form a group. Why? ( R,+, £ ) froms a commutative field. Exercise: Write down the axioms for a group, abelian group, a ring and a field. Exercise: What algrebraic structure is associated with the integers ( Z,+, £ )? Exercise: Draw a line and represent the numbers R. Mark 0, 1, 2, -1, ½, .

3. A Skew Field K A skew field is a set K endowed with two constants 0 and 1, two unary operations -: K ! K, ‘: K ! K, and with two binary operations: +: K £ K ! K, ­ : K £ K ! K, satisfying the following axioms: (x + y) + z = x + (y +z) [associativity] x + 0 = 0 + x = x [neutral element] x + (-x) = 0 [inverse] x + y = y + x [commutativity] (x ­ y) ­ z = x ­ (y ­ z). [associativity] (x ­ 1) = (1 ­ x) = x [unit] (x ­ x’) = (x’ ­ x) = 1, for x  0. [inverse] (x + y) ­ z = x ­ z + y ­ z. [left distributivity] x ­ (y + z) = x ­ y + y ­ z. [right distributivity] A (commutative) field satisfies also: x ­ y = y ­ x.

4. Examples of fields and skew fields Reals R Rational numbers Q Complex numbers C Quaterions H. (non-commutative!! Will consider briefly later!) Residues mod prime p: F p. Residues mod prime power q = p k : F q. (more complicated, need irreducible poynomials!!Will consider briefly later!)

5. Complex numbers C.  = a + bi 2 C.  * = a – bi.

6. Quaternions H. Quaternions form a non-commutative field. General form: q = x + y i + z j + w k., x,y,z,w 2 R. i 2 = j 2 = k 2 =-1. q = x + y i + z j + w k. q’ = x’ + y’ i + z’ j + w’ k. q + q’ = (x + x’) + (y + y’) i + (z + z’) j + (w + w’) k. How to define q.q’ ? i.j = k, j.k = i, k.i = j, j.i = -k, k.j = -i, i.k = -j. q.q’ = (x + y i + z j + w k)(x’ + y’ i + z’ j + w’ k) Exercise: There is only one way to complete the definition of multiplication and respect distributivity! Exercise: Represent quaternions by complex matrices (matrix addition and matrix multiplication)! Hint: q = [   ; -  *  * ].

7. Residues mod n: Z n. Two views: Z n = {0,1,..,n-1}. Define ~ on Z : x ~ y $ x = y + cn. Z n = Z /~. ( Z n,+) an abelian group, called cyclic group. Here + is taken mod n!!!

8. Example ( Z 6, +). +012345 0 0 1 2 3 4 5 1123450 2234501 3345012 4450123 5501234

9. Example ( Z 6, £ ). £ 012345 0 0 0 0 0 0 0 1012345 2024024 3030303 4042042 5054324

10. Example ( Z 6 \{0}, £ ). £ 12345 112345 224024 330303 442042 554324 It is not a group!!! For p prime, ( Z p \{0}, £ ) forms a group: ( Z p, +, £ ) = F p.

11. Vector space V over a field K +: V £ V ! V (vector addition).: K £ V ! V. (scalar multiple) (V,+) abelian group ( +  )x = x +  x. 1.x = x (  ).x = (  x)..(x +y) =.x +.y.

12. Euclidean plane E 2 and real plane R 2. R 2 = {(x,y)| x,y 2 R }. R 2 is a vector space over R. The elements of R 2 are ordered pairs of reals. (x,y) + (x’,y’) = (x+x’,y+y’) (x,y) = ( x, y). We may visualize R 2 as an Euclidean plane (with the origin O).

13. Subspaces Onedimensional (vector) subspaces are lines through the origin. (y = ax) Onedimensional affine subspaces are lines. (y = ax + b) o y = ax y = ax + b

14. Three important results Thm1: Through any pair of distinct points passes exactly one affine line. Thm2: Through any point P there is exactly one affine line l’ that is parallel to a given affine line l. Thm3: There are at least three points not on the same affine line. Note: parallel = not intersecting or identical!

15. 2. Affine Plane Axioms: A1: Through any pair of distinct points passes exactly one line. A2: Through any point P there is exactly one line l’ that is parallel to a given line l. A3: There are at least three points not on the same line. Note: parallel = not intersecting or identical!