Vector Spaces
A vector space( or linear space) V over a number field² F consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so, that for each pair of elements x, y, in V there is a unique element x + y in V, and for each element a in F and each element x in V there is a unique element ax in V, such that the following conditions hold.
- (VS 1) For all in V, (commutativity of addition).
- (VS 2) For all in V, (associativity of addition).
- (VS 3) There exists an element in V denoted by such that for each in V.
- (VS 4) For each element in V there exists an element in V such that .
- (VS 5) For each element in V, .
- (VS 6) For each pair of element in F and each element in V, .
- (VS 7) For each element in F and each pair of elements in V, .
- (VS 8) For each pair of elements in F and each pair of elements in V, .
Read more about this topic: Theorems And Definitions In Linear Algebra
Famous quotes containing the word spaces:
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)