 |
| source: adobe stock |
Let V be a non-empty set and addition is an internal binary operation denoted
by '+' that is a map from V x V → V which assigns to each ordered pair (x, y) ∈ V × V
to a unique element of V denoted by x+y and multiplication is an external
binary operation over a field F denoted ' . ' by that is a map from Fx V→ V which
assigns to each ordered pair (α, x) ∈ F × V to a unique element of V denoted
by α.x= αx, then V is called a (Left) vector space over the field F .
if the following conditions hold:
(i) (V, +) is an abelian group. The identity of the group (V, +) is called zero vector
denoted by 0.
(ii) α(x+ y) = αx+ αy for all x, y ∈V and α ∈ F
(iii) (α+ β)x = αx + βx for all α, β ∈ F and x ∈ V
(iv) α(βx) = (αβ)x = β(αx) for all α, β∈ F and x ∈ V
(v) 1 · x = x for all x ∈ V
Here 1 is the multiplicative identity of F.
0 Comments