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

Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition

Question:

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?