Chapter 4 General Vector Spaces Face Recognition ( á¹¢å ± 彐嬀)

Vector Spaces of Matrices (1/2)• Use vector notation () for the elements of M 22 (the set ofreal 2 2 matrices). Let a bu and v c d be two arbitrary 2 2 matrices. We get egfh LA 04_5Axiom 1: au v cbd e gfh a ce bg df h u + v is a 2 2 matrix. Thus M 22 is closed under addition.Axiom 3: (Theorem 2.2). a e b f u v v u c g d h Vector Spaces of Functions (1/2)LA 04_7Let V be the set of all functions () having the realnumbers as their domain ()Axiom 1: f + g is defined by (f + g)(x) = f (x) + g (x).• f + g is thus a functions with domain the set of real numbers.• f + g is an element of V closed under addition.•Example: f (x) = x and g (x) = x 2 . Then define (f + g)(x) = x + x 2Axiom 2:• c f is defined by ( c f )( x ) = c [ f(x) ].• c f is thus a functions with domain the set of real numbers.• c f is an element of V closed under scalar multiplication.• Example: Define (3 g)(x) = 3 x 2Vector Spaces of Matrices (2/2)Axiom 5: 0 0The 2 2 zero matrix is 0 , since 0 0 b 0 0 a bd 0 0 c d a b, since c d b a a b b d c c d d u 0 a c Axiom 6:If u a b c d , then uu ( u ) a b a c d c u 000 00 LA 04_6M mn , the set of m n matrices, is a vector space.Vector Spaces of Functions (2/2)Axiom 5: zero functionLet 0 be the function such that 0(x) = 0 for every real number x.(f + 0)(x) = f(x) + 0(x) = f(x) + 0 = f(x) for every real number x. f + 0 = fLA 04_8Axiom 6:[ f ( f )]( x ) f ( x ) 0 0 ( x( f )( x ) f ( x ) [ f ( x )]) f is the negative of f.• —