Basic definition of continuity
Continuity and the Axiom of Choice
Dimensionality of null space when Trace is Zero This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216).  I've succeeded in proving the first three parts, but the most interesting part of the problem eludes me.  Part (a) requires the reader to prove that $\operatorname{Tr}{(AB)} = \operatorname{Tr}{(BA)}$, which I was able to show by writing out each side of the equation using sigma notation.  Part (b) asks the reader to use part (a) to show that similar matrices have the same trace.  If $A$ and $B$ are similar, then  $\operatorname{Tr}{(A)} = \operatorname{Tr}{(S^{-1}BS)}$  $= \operatorname{Tr}(BSS^{-1})$  $= \operatorname{Tr}(B)$,  which completes part (b).  Part (c) asks the reader to show that the vector subspace of matrices with trace equal to zero have dimension $n^2 - 1$.  Curtis provides the hint that the map from $M_n(F)$ to $F$ is a linear transformation.  From this, I used the theorem that $\dim T(V) + \dim n(T) = \dim V$ to obtain the dimension of the null space.  Part (d), however, I'm stuck on.  It asks the reader to show that subspace described in part (c) is generated by matrices of the form $AB - BA$, where $A$ and $B$ are arbitrary $n \times n$ matrices.  I tried to form a basis for the subspace, but wasn't really sure what it would look like since an $n \times n$ matrix has $n^2$ entries in it, but the basis would need $n^2 - 1$ matrixes.  I also tried to think of a linear transformation whose image would have the form of $AB - BA$, but this also didn't help me.  I'm kind of stuck...  Many thanks in advance! I'm currently an M. Phil student in Math and I failed a course last semester because I was not able to hand in a home work assignment (which was worth 40% of my grade) on time (I was 30 minutes late).   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By default all users have read access, but domain policies can be set to disable access permissions for restricted accounts (and or groups). Binding against the AD has a serious overhead, the AD schema cache has to be loaded at the client (ADSI cache in the ADSI provider used by DirectoryServices). This is both network, and AD server, resource consuming - and is too expensive for a simple operation like authenticating a user account. You're relying on an exception failure for a non-exceptional case, and assuming that means invalid username and password. Other problems (e.g. network failure, AD connectivity failure, memory allocation error, etc) are then mis-intrepreted as authentication failure.   Method 2. LogonUser Win32 API   have suggested using the  API function. This sounds nice, but unfortunately the calling user sometimes needs a permission usually only given to the operating system itself:     The process calling LogonUser requires   the SE_TCB_NAME privilege. If the   calling process does not have this   privilege, LogonUser fails and   GetLastError returns   ERROR_PRIVILEGE_NOT_HELD.       In some   cases, the process that calls   LogonUser must also have the   SE_CHANGE_NOTIFY_NAME privilege   enabled; otherwise, LogonUser fails   and GetLastError returns   ERROR_ACCESS_DENIED. This privilege is   not required for the local system   account or accounts that are members   of the administrators group. By   default, SE_CHANGE_NOTIFY_NAME is   enabled for all users, but some   administrators may disable it for   everyone.   Handing out the "Act as a part of the operating system" privilege is not something you want to do willy-nilly - as Microsoft points out in a :     ...the process that is calling   LogonUser must have the SE_TCB_NAME   privilege (in User Manager, this is   the "Act as part of the Operating   System" right). 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