The trace of a square matrix is
the sum of the entries on
its main diagonal.
Let be the vector space of all matrices with real entries.
Let be the set of all matrices with real entries that have trace .
Is a subspace of the vector space ?
- Does contain the zero vector of ?
- Is closed under addition? If it is, enter CLOSED.
If it is not, enter two matrices in whose sum is not in ,
using a comma separated list and syntax such as
for the answer .
(Hint: to show that is not closed under addition, it is sufficient to find two trace one
matrices and such that has trace not equal to one.)
- Is closed under scalar multiplication? If it is, enter CLOSED.
If it is not, enter a scalar in and a matrix in whose product is not in ,
using a comma separated list and syntax such as for the
answer .
(Hint: to show that is not closed under scalar multiplication, it is sufficient to find a real number
and a trace one matrix such that has trace not equal to one.)
- Is a subspace of the vector space ?
You should be able to justify your answer by writing a complete, coherent, and detailed proof
based on your answers to parts 1-3.
In order to get credit for this problem all answers must be correct.