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Let S be the set of all ordered pairs of real numbers. De?ne scalar multiplication and...

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Advance Math

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.

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