Vector Space
Published On :2021-04-27 05:36:00
Definition : A vector space is a nonempty set V with two operation (addition, scalar, multiplication ) that satisfy the following conditions:
[begin{array}{l}
1)u + v;belongs;V;;;;;;whenever;u,v;belongs;V\\
2)u + v = v + u;;;;;;for;all;;u,v;belongs;V\\
3)left( {u + v} right) + w = u + left( {v + w} right);;;for;all;u,v,w;belongs;V\\
4);There;;exists;{0_v}belongs;V;such;that;{0_v} + u = u\\
5)for;all;u;belongs;V;,;there;exists;left( { - u} right)belongs;V;;such;that;u + left( { - u} right) = {0_v}\\
6)ku;belongs;V;;for;all;k;belongs;F;;and;u;belongs;V;left( {F = reals;Or;complex} right)\\
7)kleft( {u + v} right) = ku + kv;,;k;belongs;F;,;u,v;belongs;V.\\
8)left( {{k_1} + {k_2}} right)u = {k_1}u + {k_2}u;;;,;{k_1},{k_2};belongs;F;,;u;belongs;V\\
9){k_1}left( {{k_2}u} right) = left( {{k_1}{k_2}} right)u\\
10)1u = u\
end{array}]
