no, the matrix you would up with certainly is still a 2x2 matrix.
and, in fact, it IS true that r.(A+B) = r(A+B)^t = r(A^t + B^t) = r.A + r.B.
how about checking if r.(s.A) = (rs).A....?
Let the set of all 2 x 2 matrices. Let the operation of vector addition in V be the usual matrix addition but let scalar multiplication in V be defined by:
Show that V is not a vector space by demonstrating with an example that one of the properties in the definition of a vector space fails to hold.
My attempt:
property I believe fails to hold (let r be any real number):
I'll define both A and B as the 2 x 2 matrices:
and
=> => =>
Hence, V is not a vector space since it does not hold for the distributivity of scalar multiplication with respect to vector addition. Did I go about this proof correctly?
OK:
While:
Clearly, and therefore, V is not a vector space. Correct?
Also, a general question: How would you approach such a question in a circumstance where you are pressed for time? I mean, in an exam, it is quite time consuming to check through all eight properties of a vector space to determine whether a set is a vector space or not. Any general or intuitive tips that you could give? Thanks in advance.