1. Vector Space

24. February 2004

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, .

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.

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!)

Complex numbers C.

• = a + bi 2 C.

• * = a – bi.

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 = [ ; -* *].

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!!!

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

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

Example (Z6\{0}, £).

£ 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

It is not a group!!!

For p prime, (Zp\{0}, £) forms a group: (Zp, +,£) = Fp.

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.

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’) = (x+x’,y+y’)

• (x,y) = ( x, y).

• We may visualize R2 as an Euclidean plane (with the origin O).

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

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!

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!