Q2(b) of HW1 reads:
"Find a subset of R^3 which is closed under vector addition,
but not scalar multiplication".
One of you suggested that one possible answer would be "all
of R^3, but with _complex_ scalar multiplication instead of
real scalar multiplication". This is, technically,
a correct (and clever) answer to the question - R^3 is indeed
closed under vector addition, and real scalar multiplication,
but not complex scalar multiplication (for instance, (1,2,3)
is a vector in R^3, but i(1,2,3) = (i, 2i, 3i) is not). However,
it is not what the question intended - I wanted you to use the
standard R^3 vector addition and standard R^3 scalar multiplication
(i.e. the real scalar multiplication). And no, you can't use
the non-standard scalar multiplication rule from Q1 for this
question either.
More generally in the HW and in the exams, when I refer to a standard
space such as R^3, or P_n, or M_{n x n}, etc., I always understand
that these spaces are using the _standard_ vector addition rule
and _standard_ scalar multiplication, unless otherwise specified.
In particular, scalar multiplication will almost always refer
to real scalar multiplication, not complex. (If you do try to
actively look for loopholes in the wording, you will probably get
some partial credit just for cleverness, but to get full credit
you should try to answer the question in the spirit it was intended).
Terry