Vectors

1. Vectors

2. Two Operations • Many familiar sets have two operations. • Sets: Z, R, 2  2 real matrices M2(R) • Addition and multiplication • All are groups under addition. • 0 is a problem for multiplication. • Without 0 the sets are groups • Multiplication is not commutative for M2(R)

3. 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 M2(R) - not a commutative ring Rings and Fields

4. The group of complex numbers (C, +) are isomorphic to (R2, +). R2 = {(a, b): a, bR} Map: a+bi (a, b) (C, +) is commutative Multiplication is defined on C (or R2). (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/a2+b2, -b/a2+b2) Complex Field

5. A vector space combines a group and a field. (V, +) a commutative group (F, +, ) a field Elements in V are vectors matrices, polynomials, functions Elements in F are scalars reals, 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 Vector Space S1

6. Cartesian Vector • A real Cartesian vector is made from a Cartesian product of the real numbers. • EN = {(x1, …, xN): xi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”. S1 x2 (x1, x2) x1

7. An algebra is a linear vector space with vector multiplication. Algebra definitions: v,w,x V Closure: v□w V 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-1v = vv-1 = 1, v V Commutative:v□w = w□v Anticommutative:v□w=-w□v Algebra

8. Quaternions • Define a group with addition on R4. • Q = {(a1, a2, a3, a4): aiR} • Commutative group • Define multiplication of 1, i, j, k by the table at left. • Multiplication is not commutative. S1 The quaternions are not isomorphic to the cyclic 4-group. next