Question

VECTOR SPACES LINEAR ALGEBRA

Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on u = (u1, u2) and v = (v1, v2): u + v = (u1 + v1 + 1, u2 + v2 + 1), ku = (ku1, ku2)

a) Show that (0,0) does not = 0

b) Show that (-1, -1) = 0

c) Show that axiom 5 holds by producing an ordered pair -u such that u + (-u) = 0 for (u1, u2)  

d) Find two vector space axioms that fail to hold

e) Compute u + v and ku for u = (0, 4), v = (1, −3), and k = 2.

Answer 1

for any u-lu 2) thon T(0.0) we hav.e But uro- (unu) (,0) (u,u (O,0) does not (b) Now (-1s-1) Laua (U19 U2) + C-1,-)- (u,-1+1, U2-11) ch (uu2) 24 (-1,-1)0 (c) uzlu,,u2) -(041-2,-42-2) thon thin utl-u) ujuz)(u2,-Uz -2). (u,-41-211 u2- 42-2 11)

(4) The atiom hold does not K(utv)uutu Belaure (ur) u,),u22)) But AlYo (MITK2)ut uu2u So atiomo (ui2)u does not ha d 2K14+4, u u l0,4) - (,-3) 2 (0,4)(1,3) (OTII,4-3 (2,2) utv (2,2) (0.4) -(2.0, 2.4) (0,8) Ku 2 and (or8) KUn