Chapter 2 – Properties of Real Numbers 2.2 – Addition of Real Numbers.
1. Vector Space 24. February 2004. Real Numbers R. Let us review the structure of the set of real...
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Transcript of 1. Vector Space 24. February 2004. Real Numbers R. Let us review the structure of the set of real...
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!