## Re: space, subspace...

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• ... In your Problem 2, you re dealing with the trace operator tr, which maps a (complex) matrix to a complex number. Its null space is the set of matrices A
Message 1 of 2 , Jun 1, 2002
--- In mathforfun@y..., "sabujakash" <sabujakash@y...> wrote:
> hi,
> What is the exact defn. of Nullspace of a operator..
> And what is a subspace?? Is it just like subsets?
> Infact what is the difference between a space and a set?
>
> Hope somebody will give me a clear idea...
> thnx

In your Problem 2, you're dealing with the trace operator tr, which
maps a (complex) matrix to a complex number. Its null space is the
set of matrices A such that tr(A) = 0, i.e. the set of all matrices
with zero trace. In general, the null space of an operator is the set
of vectors which are mapped to zero.

The word "space" is somewhat ambiguous and could have many meanings in
mathematics, but judging from the context of your questions we are
talking about the set of n by n complex matrices, considered as a
vector space over the field of complex numbers. A vector space is a
set of elements, called "vectors", with two operations defined on
them-vector addition and scalar multiplication-which satisfy a certain
set of axioms (about eleven axioms in all, if I recall correctly). If
you're not familiar with the axioms for a vector space, you can find
them in a book on linear algebra, or if you ask again I'm sure someone
will post them for you. A subspace is more than just a set; it is a
subset of the vector space which is a vector space in its own right.
There is a fundamental theorem of linear algebra which states that a
non-empty subset of a vector space is a subspace if it is closed under
the operations of addition and scalar multiplication. I.e., if S is
the subset and x and y are members of S, then x+y is in S; and if c is
any scalar then cx is in S. Another fundamental theorem states that
the null space of a linear transformation (for example, tr) is always
a subspace. What Jason was asking you about earlier (I think) was
whether Problem 2 is correctly stated, since it is not immediately
clear that {A: A = BC-CB for some B, C} is in fact a subspace. In
particular, it's not obvious (to me, at least) that it is closed under
addition. On the other hand, it's not obvious that it's not closed,
either; so maybe it's a subspace after all.

If you don't mind a word of advice, your set of 11 problems is pretty
ambitious if you are not sure of the definition of a vector space-at
least, most of them look pretty hard to me, and I've known what a
vector space is for a pretty long time. Long enough to forget a lot
of stuff, anyway. ;-)
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