Let V be the set of all ordered triples of real numbers. For u = (u1,...

Let V be the set of all ordered triples of real numbers. For u = (u1, u2, u3) and v = (v1, v2, v3), we define the following operations of addition and scalar multiplication on V :

u + v = (u1 + v1, u2 + v2 − 1, u3 + v3 − 2) and ku = (ku1, ku2, ku3).

For example, if u = (1, 0, 3), v = (2, 1, 1), and k = 2 then

u + v = (1 + 2, 0 + 1 − 1, 3 + 1 − 2) = (3, 0, 2) and 2u = (2 · 1, 2 · 0, 2 · 3) = (2, 0, 6).

Complete the following:

(a) Calculate (1, 1, 1) + (2, 2, 2).

(b) Show that (0, 0, 0) 6= 0.

(c) What is 0?

(d) State a vector space axiom that fails to hold. Give an example to justify your claim.

Expert Solution

milcah answered 2 years ago

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