1. Vector Space

1
1. Vector Space

• 3. March 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,,) forms a (commutative) field.
• Exercise Write down the axioms for a group,
  abelian group, a ring and a field.
• Exercise What algebraic 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 2 C.
• a a bi.

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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 a b -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!!!

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Example (Z6, ).
0 1 2 3 4 5
0 0 1 2 3 4 5
1 1 2 3 4 5 0
2 2 3 4 5 0 1
3 3 4 5 0 1 2
4 4 5 0 1 2 3
5 5 0 1 2 3 4
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Example (Z6, ).
0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 0 2 4
3 0 3 0 3 0 3
4 0 4 2 0 4 2
5 0 5 4 3 2 4
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Example (Z6\0, ).
It is not a group!!! For p prime, (Zp\0, )
forms a group (Zp, ,) Fp.
1 2 3 4 5
1 1 2 3 4 5
2 2 4 0 2 4
3 3 0 3 0 3
4 4 2 0 4 2
5 5 4 3 2 4
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Vector space V over a field K

• V V ! V (vector addition)
• . K V ! V. (scalar multiple)
• (V,) abelian group
• (l m)x l x m x.
• 1.x x
• (l m).x l(m x).
• l.(x y) l.x l.y.

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Euclidean plane E2 and real plane R2.

• R2 (x,y) x,y 2 R.
• R2 is a vector space over R. The elements of R2
  are ordered pairs of reals.
• (x,y) (x,y) (xx,yy)
• l(x,y) (l x,l y).
• We may visualize R2 as an Euclidean plane (with
  the origin O).

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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
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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!

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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!

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Examples

• Each affine plane is an incidence structure C
  (P,L,I) of points and lines.
• Let K be a field, then K2 has a structure of an
  affine plane.
• K Fp.
• Determine the number of points and lines in the
  affine plane A2(p) Fp2.

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Parallel Lines

• Parallel lines l m define an equivalence
  relation on the set of lines.
• l l
• l m ) m l
• l m, m n ) l n.
• Conditions 1. and 2. are true for any incidence
  structure. (Prove it!)
• Exercise Prove condition 3 for affine planes and
  find a counter-example for general incidence
  structure.

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A bundle of parallel lines

• An equivalence class of parallel lines is called
  a bundle of parallel lines.
• Thm. Each bundle of parallel lines defines an
  equivalence relation on the set of points.

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Ideal points and Ideal line

• Each bundle of parallel lines defines a new
  point, called an ideal point (or a point at
  inifinity.) New point is incident with each line
  of the bundle.
• In addition we add a new ideal line (or line at
  infinity)

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Extended Plane

• Let A be an arbitrary affine plane. The incidence
  structure obtained from A by adding ideal points
  and ideal lines is celled the extended plane and
  is denoted by P(A).

• Theorem. Let C be an extended plane obtainde from
  any affine plane. The following holds
• T1. For any two distinct points P and Q there
  exists a unique line l connecting them.
• T2. For any two distinct line l and m there
  exists a unique point P in their intersesction.
• T3. There exist at least four points P,Q,R,S such
  that no three of them are colinear.

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Projective Plane

• Axioms for the Projective Plane. Let C be an
  incidence structure of points and lines that
  satisfies the following axioms
• P1. For any two distinct points P and Q there
  exists a unique line l connecting them.
• P2. For any two distinct line l and m there
  exists a unique point P in their intersesction.
• P3. There exist at least four points P,Q,R,S such
  that no three of them are colinear.

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Linear Transformations

• In a vector space the important mappings are
  linear transformations
• L(l x m y) l L(x) m L(y). L-1 exists.
• L can be represented by a nonsingular square
  matrix.

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Semi Linear Transformations

• A semi linear transformation is more general
• L(lx m y) f(l) L(x) f(m) L(y). L-1 exists,
  f K ! K is an automorphism of K.

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Exercise

• Prove that in R, Q, Fp, (p- prime) there are no
  nontrivial automorphisms.
• Prove that z a z (conjugate) is an automorphism
  of C.

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Affine Transformations

• In an affine plane the important mappings are
  affine transformations (affinities).
• An affine transformation maps three collinear
  points to collinear points.
• Each affine transformation is of the form
• A(x) c, where A is a semilinear transformation.