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!
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