Field • A field F is a set together with 2 binary operations • +, × (− optional) that satisfies the following: ○ a+b=b+a ○ (a+b)+c=a+(b+c) ○ a×b=b×a ○ (a×b)×c=a×(b×c) ○ a×(b+c)=a×b+a×c ○ There is a special element O, such that a+O=a ○ There is a special element 1, such that 1×a=a ○ For all a, there is a b, such that a+b=0 ○ For any a≠O, there is a b, such that a×b=1 ○ Optional:1≠O, O≠1 • Example ○ F={0,1} ○ +≔{█(0+0=0@0+1=1@1+1=0)┤ ○ ×≔{█(0×0=0@0×1=0@1×1=1)┤ • Example ○ F={0,1,2} ○ +≔{█(0+0=0@0+1=1@0+2=2@1+1=2@1+2=0@2+2=1)┤ ○ ×≔{█(0×0=0@0×1=0@0×2=0@1×1=1@1×2=2@2×2=1)┤ Vector Space • A vector space V(over F) is a set together with binary operations • {█(+:V+V→V@×:F×V→V)┤, such that ○ F is a field ○ u+v=v+u, ∀u,v∈V ○ (u+v)+w=v+(u+w), ∀u,v,w∈V ○ There is a 0 and vector 0 ⃗, such that § ∀u,v∈V, ∀a,b∈F § u+0 ⃗=u § 0×u=0 ⃗ § a×0 ⃗=0 ⃗ § (a×b)×u=a×(b×u) § (a+b)×u=a×u+b×u § a(u+v)=a×u+a×v § u+(−1)u=(1+(−1))×u=0×u=0 ⃗
