12-240/Classnotes for Tuesday September 25

Today's class dealt with the properties of vector spaces.

Definition

Let F be a field. A vector space V over F is a set V of vectors with special element O ( of V) and two operations: (+): VxV->V, (.): FxV->V

VxV={(v,w): v,w ${\displaystyle \in \!\,}$ V}

FxV={(c,v): c ${\displaystyle \in \!\,}$ F, v ${\displaystyle \in \!\,}$ V}

Then, (+): VxV->V is (v,w)= v+w; (.): FxV->V is (c,v)=cv

Such that

VS1 ${\displaystyle \forall \!\,}$ x, y ${\displaystyle \in \!\,}$ V: x+y = y+x

VS2 ${\displaystyle \forall \!\,}$ x, y, z ${\displaystyle \in \!\,}$ V: x+(y+z) = (x+y)+z

VS3 ${\displaystyle \forall \!\,}$ x ${\displaystyle \in \!\,}$ V: 0 ( of V) +x = x

VS4 ${\displaystyle \forall \!\,}$ x ${\displaystyle \in \!\,}$ V, ${\displaystyle \exists \!\,}$ V ${\displaystyle \in \!\,}$ V: v + x= 0 ( of V)

VS5 ${\displaystyle \forall \!\,}$ x ${\displaystyle \in \!\,}$ V, 1 (of F) .x = x

VS6 ${\displaystyle \forall \!\,}$ a, b ${\displaystyle \in \!\,}$ F, ${\displaystyle \forall \!\,}$ x ${\displaystyle \in \!\,}$ V: (ab)x = a(bx)

VS7 ${\displaystyle \forall \!\,}$ a ${\displaystyle \in \!\,}$ F, ${\displaystyle \forall \!\,}$ x, y ${\displaystyle \in \!\,}$ V: a(x + y)= ax + ay

VS8 ${\displaystyle \forall \!\,}$ a, b ${\displaystyle \in \!\,}$ F, ${\displaystyle \forall \!\,}$ x ${\displaystyle \in \!\,}$ V: (a + b)x = ax + bx

Polynomials

Definition : Pn(F) = {all polynomials of degree less than or equal to n with coefficients in F}

                        = {anx^n + an-1x^n-1 + ... + a1x^1 + a1x^1}


0 = 0x^n + 0x^n-1 +...+ 0x^0

addition and multiplication: as you imagine

P(f) = {all polynomials with coefficients in F}

Take F= Z/2 |F| = 2

|P(F)| = infinite

in Pn(Z/2) x^3≠x^2

                  x^3 = 1*x^3+0x^2+0x+O = f
x^2 = 1*x^2+0x+0 = g
yet f(0)= g(0) and f(1)=g(1)


Theorem

1. Cancellation Laws

  (a) x+z=y+z         ==> x=y
(b) ax=ay,a≠0       ==> x=y
(c) x≠0 of V, ax=bx ==> a=b


2. 0 of V is unique

3. Negatives are unique (so subtraction makes sense

4.(0 of F)x = 0 of V

5. a∙0=0

6. (-a)x = -(ax) = a(-x)

7. a∙v=0 <==> a=0 or v=0

Proof

1. (a) x+z=y+z

       Find a w s.t. z+w=0 (V.S. 4)
(x+z)+w = (y+z)+w
Use VS2
x+(z+w) = y +(z+w)
x + 0   = y + o
Use VS3      x=y