Question: Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on 5 by (x1, x2) = (x1, x1,

Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on 5 by
α(x1, x2) = (αx1, αx1, αx2)
(x1, x2) ⊕ (y1, y2) = (x1 + y1, 0)
We use the symbol ⊕ to denote the addition operation for this system to avoid confusion with the usual addition x + y of row vectors. Show that S, with the ordinary scalar multiplication and addition operation ⊕, is not a vector space. Which of the eight axioms fail to hold?

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