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AM 106/206: Applied Algebra Prof. Salil Vadhan

Lecture Notes 20

November 22, 2010

1 Vector Spaces

•   Reading: Gallian Ch. 19

•  Today’s main message: linear algebra (as in Math 21) can be done over any field, and mostof the results you’re familiar with from the case of  R  or  C  carry over.

•   Def:   A  vector space  over a field  F   is a set  V  with two operations + :  V  × V   →  V   (vectoraddition) and · :  F  × V  → V   (scalar multiplications) that satisfy the following properties:

1.   V   is an abelian group under +.2. (ab) · v =  a · (b · v) for all  a, b ∈  F   and  v ∈  V  .

3. 1 · v =  v  for all  v ∈  V  .

4.   a · (v + w) = a · v + a · w  for all  a  ∈  F   and  v , w ∈  V  .

5. (a + b) · v =  a · v + b · v  for all  a, b ∈  F   and  v  ∈  V  .

•   A vector space has more structure than an abelian group, but less structure than a ring (onlymultiplication by scalars, not multiplication of arbitrary pairs of elements of  V  ).

•  Examples and Nonexamples:

–   V   = F n

–   V   = C,  F   = R

–   V   = Zn,  F  = Z2

–   V   = F [x]

–   V   = F [x]/ p(x)

–   V   = R  for a ring R  containing  F .

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•   Prop:   Every  n-dimensional vector space  V   over  F   is isomorphic to  F n.

Proof:   Let v1, . . . , vn  be a basis for  V  .Then an isomorphism from  F n to  V   is given by:

•   Matrices:  A linear map  f   :  F n →  F m can be described uniquely by an  m × n  matrix  M with entries from F .

–   M ij  = f (e j)i, where  e j  = (000 · · · 010 · · · 00) has a 1 in the  j ’th position.

–   For   v   = (v1, . . . , vn)  ∈  F n,   f (v)i   =  f (

 j v je j)i   =

 j v jf (e j)i   =

i M ijv j   = (M v)i,where M v   is matrix-vector product.

–   Matrix multiplication  ↔  composition of linear maps.

–   If  n  =  m, then  f  is an isomorphism  ⇔  det(M ) = 0.

–   Solving  M v   =  w   for  v  (when given   M   and  w  ∈  F m) is equivalent to solving a linearsystem with  m  variables and  n unknowns.

•   Example:   f   : Z33 → Z3

2  given by  f (v1, v2, v3) = (v1 + 2v2, 2v1 + v3).

•   Thm:   If  f   : V  → W   is a linear map, then dim(ker(f )) + dim(im(f )) = dim(V  ).

Proof:   omitted.

•   When F  finite, this says  |V | =  |F |dim(V  ) = |F |dim(ker(f )) · |F |dim(im(f )) = | ker(f )| · |im(f )|, justlike for group homomorphisms!

•   Corollaries:

–   F n F m if  m = n.

–  All bases of a vector space have the same size.

–  A homogenous linear system  M v = 0 for a given  m × n matrix M  always has a nonzerosolution  v   if  n > m  (more variables than unknowns).

•   Computational issues:   For  n × n  matrices over  F ,

–  Matrix multiplication can be done with O(n3) operations in  F  using the standard algo-rithm.

–  The determinant and inverse, and solving a linear system  M v  =  w  can be done usingO(n3) operations in  F   using Gaussian elimination. (For infinite fields, need to worryabout the size of the numbers, or accuracy if doing approximate arithmetic. No suchproblem in finite fields.)

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–  Asymptotically fastest known algorithms run in time  O(n2.376). Whether time  O(n2) ispossible is a long-standing open problem.

3 Application to Extension Fields

•   Reading: parts of Gallian Ch. 21

•   Def:   E   is an  extension field   of   F   is   F   is a subfield of   E . The   degree of   E   over   F   is thedimension of  E  as a vector space over  F , and is denoted [E   :  F ].   E   is a   finite extension   if [E   : F ] is finite.

•   Examples:

–   [C : R] =

–   [F [x]/ p(x) :  F ] =

–   [F(α) : F] =

•   Thm 21.5:   If  K   is a finite extension of  E , and  E   is a finite extension of  F , then [K   : F ] =[K   : E ][E  :  F ].

•   Example:  The splitting field of  x8 − 1 over  Q, i.e.  Q(ω) = Q(i)(ω), where  ω  =  e2πi/8.

•   Proof:

•   Corollary:   If   E   is a finite extension of  F   and   α  ∈  E , then  α   is algebraic over  F   and itsminimal polynomial has degree dividing [E   : F ].

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