Spatii vectoriale

Spatii vectoriale

Fie un camp (corp comutativ). In V se definesc:

→ o lege de comp. interna + : V V → V

adunare

(x,y) → x + y

→ o lege de comp. externa : K V → V

multiplicare cu scalari

(α,x) → α ∙ x

Def. 1

Multimea nevida V se numeste principalul vector peste K daca:

I. (V, +) - grup abelian

a) (x, y, z) V (x + y) + z = x + (y + z) (asoc.)

b) 0 V a. i. x + 0 = 0 + x , x V (el. neutru)

c) x' a. i. x + x' = x' + x = 0 ,x', x V (el. simetric)

d) x, y V , x + y = y + x (comut.)

II. multiplicarea cu scalari verifica

e) α (x + y) = αx + αy , α K; x,y V R: R

f) (α + β) x = αx + βx , α, β K; x V x = (x, x)

g) α (βx) = (αβ) x

h) 1∙x = x , x V

R: x = (x, x, x)

R: x + y = (x, x) + (y, y) = (x+ y, x+ y)

Ex: (3, -4) + (2, 0) = (3 + 2, -4 + 0) = (5, -4)

α x = α (x, x) = (α x, α x)

Ex: 3 (1, -4) = (3 ∙ 1, 3 ∙ (-4)) = (3, -12)