Multivariable Advanced Calculus

3.1. ALGEBRA IN F N , VECTOR SPACES 253.1 Algebra in F n , Vector SpacesThere are two algebraic operations done with elements of F n . One is addition and theother is multiplication by numbers, called scalars. In the case of C n the scalars arecomplex numbers while in the case of R n the only allowed scalars are real numbers.Thus, the scalars always come from F in either case.Definition 3.1.1 If x ∈ F n and a ∈ F, also called a scalar, then ax ∈ F n isdefined byax = a (x 1 , · · · , x n ) ≡ (ax 1 , · · · , ax n ) . (3.1)This is known as scalar multiplication. If x, y ∈ F n then x + y ∈ F n and is defined byx + y = (x 1 , · · · , x n ) + (y 1 , · · · , y n )the points in F n are also referred to as vectors.≡ (x 1 + y 1 , · · · , x n + y n ) (3.2)With this definition, the algebraic properties satisfy the conclusions of the followingtheorem. These conclusions are called the vector space axioms. Any time you have aset and a field of scalars satisfying the axioms of the following theorem, it is called avector space.Theorem 3.1.2 For v, w ∈ F n and α, β scalars, (real numbers), the followinghold.v + w = w + v, (3.3)the commutative law of addition,(v + w) + z = v+ (w + z) , (3.4)the associative law for addition,the existence of an additive identity,v + 0 = v, (3.5)v+ (−v) = 0, (3.6)the existence of an additive inverse, Alsoα (v + w) = αv+αw, (3.7)(α + β) v =αv+βv, (3.8)α (βv) = αβ (v) , (3.9)1v = v. (3.10)In the above 0 = (0, · · · , 0).You should verify these properties all hold. For example, consider 3.7α (v + w) = α (v 1 + w 1 , · · · , v n + w n )= (α (v 1 + w 1 ) , · · · , α (v n + w n ))= (αv 1 + αw 1 , · · · , αv n + αw n )= (αv 1 , · · · , αv n ) + (αw 1 , · · · , αw n )= αv + αw.As usual subtraction is defined as x − y ≡ x+ (−y) .