Scalar+Multiplication

Scalar Multiplication: multiplication of a scalar by a vector.

Scalar multiplication definition has some rules to obey: 1. Associability which says: (ac)b = a(cb), where a is function in K and b in V.

2. When we multiply a vector by 1, it does not change: 1.a = a.

3. But when we multiply a vector by 0, then it gives a null vector: 0.a = 0.

4. Left distributivity says: ( a + c )b = ab + cb.

5. Right distributivity says : a( b + c ) = ab + ac

6. When we multiply a vector by -1, then it gives the additive inverse of it: (-1)a = -a. Here addition can either be in field or vector space. Scalar multiplication can also be described like an action of fields or external binary operations are occurring between the fields on the vector space. Scalar multiplication definition can also be analyzed geometrically as shrinking or stretching of vectors. It can also be done by the simple multiplication in the field, when V is taken to be K and this case is a special case. But when V is taken to be K, then this type of scalar multiplication is termed as component-wise.

Explanation Video of Scalar Multiplication: