Vectors

Two Operations  Many familiar sets have two operations. Sets: Z, R, 2  2 real matrices M 2 ( R )Sets: Z, R, 2  2 real matrices M 2 ( R ) Addition and multiplicationAddition and multiplication All are groups under addition.All are groups under addition.  0 is a problem for multiplication. Without 0 the sets are groupsWithout 0 the sets are groups Multiplication is not commutative for M 2 ( R )Multiplication is not commutative for M 2 ( R )

Rings and Fields  A ring is a set with two operations (R, +,  )  Rings are commutative groups under addition.  Multiplication is associative and distributive.  A field is a commutative ring with multiplicative identity and inverses for all except 0. Question  Are there rings which are not fields?  Examples: Z - no multiplicative inverses M 2 ( R ) - not a commutative ring

Complex Field  The group of complex numbers ( C, +) are isomorphic to ( R 2, +). R 2 = {( a, b ): a, b  R }R 2 = {( a, b ): a, b  R } Map: a+bi  ( a, b )Map: a+bi  ( a, b ) (C, +) is commutative(C, +) is commutative  Multiplication is defined on C (or R 2 ). ( a, b )  ( c, d ) = (ac-bd, ad+bc)( a, b )  ( c, d ) = (ac-bd, ad+bc)  Prove ( C, +,  ) is a field.  Multiplication on C is commutative. ( a, b )  ( c, d ) = (ac-bd, ad+bc) = (ca-db, da+cb) = ( c, d )  ( a, b )  The multiplicative identity is (1, 0).  Every non-zero element has an inverse. (a, b) -1 = (a/a 2 +b 2, -b/a 2 +b 2 )

Vector Space  A vector space combines a group and a field. (V, +) a commutative group(V, +) a commutative group (F, +,  ) a field(F, +,  ) a field  Elements in V are vectors matrices, polynomials, functionsmatrices, polynomials, functions  Elements in F are scalars reals, complex numbersreals, complex numbers  Scalar multiplication provides the combination. v, u  V ; f, g  F  Closure: fv  V  Identity: 1v = v  Associative: f(gv) = (fg)v  Distributive: f(v+u) = fv + fu (f+g)v = fv + gv S1S1

Cartesian Vector  A real Cartesian vector is made from a Cartesian product of the real numbers. E N = {( x 1, …, x N ): x i  R } Addition by component Multiplication on each component  This specific type of a vector is what we think of as having a “magnitude and direction”. S1S1 x1x1 x2x2 (x 1, x 2 )

Algebra  An algebra is a linear vector space with vector multiplication.  Algebra definitions: v,w,x  V Closure: v□w  VClosure: v□w  V Bilinearity:Bilinearity: ( v+w)□x = v□x + w□x x□ ( v+w) = x□v + x□w  Some algebras have additional properties. Associative: v □ ( w□x ) = ( v□w ) □x Identity:  1  V, 1v = v1 = v,  v  V Inverse:  v -1  V, v -1 v = vv -1 = 1,  v  V Commutative: v□w = w□v Anticommutative: v□w =  w□v

Quaternions  Define a group with addition on R 4. Q = {( a 1, a 2, a 3, a 4 ): a i  R } Commutative group  Define multiplication of 1, i, j, k by the table at left.  Multiplication is not commutative. S1S1 The quaternions are not isomorphic to the cyclic 4-group. next