1. Vector Space

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, p.

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 ¹ 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: Fp. • Residues mod prime power q = pk: Fq. (more complicated, need irreducible poynomials!!Will consider briefly later!)

5. Complex numbers C. • a = a + bi 2C. • a* = 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 2R. • 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 = [ab; -b*a*].

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

8. Example (Z6, +).

9. Example (Z6, £).

10. Example (Z6\{0}, £). It is not a group!!! For p prime, (Zp\{0}, £) forms a group: (Zp, +,£) = Fp.

11. Vector space V over a field K • +: V £ V ! V (vector addition) • .: K £ V ! V. (scalar multiple) • (V,+) abelian group • (l + m)x = lx + mx. • 1.x = x • (lm).x = l(mx). • l.(x +y) = l.x + l.y.

12. Euclidean plane E2 and real plane R2. • R2 = {(x,y)| x,y 2R}. • R2 is a vector space over R. The elements of R2 are ordered pairs of reals. • (x,y) + (x’,y’) = (x+x’,y+y’) • l(x,y) = (l x,l y). • We may visualize R2 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) y = ax + b y = ax o

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!