Question

Define an operation of scalar multiplication in vector space

Answer 1

Definition: A vector space is a set V on which two operations + and · are defined, called vector addition and scalar multiplication.

The operation + (vector addition) must satisfy the following conditions:

Closure: If u and v are any vectors in V, then the sum   u + v   belongs to V.

(1) Commutative law: For all vectors u and v in V,     u + v = v + u

(2) Associative law: For all vectors u, v, win V,     u + (v + w) = (u + v) + w

(3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V,     0 + v = v  and   v + 0 = v.

(4) Additive inverses: For each vector v in V, the equations     v + x = 0   and   x + v = 0     have a solution x in V, called an additive inverse of v, and denoted by - v.

Answer 2

here is question's answer


Definition: A vector space is a set V on which two operations + and · are defined, called vector addition and scalar multiplication.
There are many conditions-
•Closure: If u and v are any vectors in V, then the sum   u + v   belongs to V.

(1) Commutative law: For all vectors uand v in V,     u + v = v + u

(2) Associative law: For all vectors u, v, win V,     u + (v + w) = (u + v) + w

(3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V,     0 + v= v   and   v + 0 = v.

(4) Additive inverses: For each vector v in V, the equations     v + x = 0   and   x + v = 0     have a solution x in V, called an additive inverse of v, and denoted by - v