Let S be the set of all ordered pairs of real numbers. De?nescalar multiplication and addition on S by:
?(x1,x2)=(?x1,?x2)
(x1,x2)?(y1,y2)=(x1 +y1,0)
We use the symbol?to denote the addition operation for thissystem in order to avoid confusion with the usual addition x+y ofrow vectors.
Show that S, together with the ordinary scalar multiplicationand the addition operation?, is not a vector space.
Test ALL of the eight axioms and report which axiomsfail to hold.
