How do I show that [(k)(u)]*v=k(u*v)?
Where k is a real number, and u and v are vectors.
- mcbengtLv 79 years agoFavorite Answer
I'm assuming * is the dot product. I will write w_j for the jth entry of the vector w, and assume that u and v have the same length n. You have
(ku)*v = the sum, from j = 1 to n, of (ku)_j v_j [this is the definition of the dot product]
= the sum, from j = 1 to n, of (k u_j) v_j [from the definition of scalar multiplication, the jth entry of ku is k times the jth entry of u]
= k (the sum, from j = 1 to n, of u_j v_j) [this is the distributive law for addition and multiplication of real numbers: a single number k can be pulled out of a sum]
= k (u * v) [from the definition of the dot product again]
So really the law (ku)*v = k(u*v) comes out of the fact that both (ku)*v and u*v are defined to be sums, and when you write out those sums, you find that the distributive law (for usual multiplication of numbers, across a sum of numbers) tells you that one is k times the other