1) Consider the following attempt to define a vector space that does not work. Let V be
the set R2 with the standard vector addition and an unusual scalar multiplication defined
by
(a, b) + (c, d) = (a + c, b + d)
k ๏(a, b) = (k 2a, k 2b) .
(The funny symbol for scalar multiplication is meant to emphasize that it’s not the
standard definition.) It turns out that all axioms hold except for axiom #8.
a) Prove that axiom #9 holds for any scalars k and m and any vector (a, b) .
b) Show that axiom #8 does not hold in general by finding particular scalars k and m
and a particular vector (a, b) that

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