For trace: there is a perspective that a trace is a generalized dimension counting operator. If we have a projection matrix (a matrix such that AA = A), then th

For trace: there is a perspective that a trace is a generalized dimension counting operator. If we have a projection matrix (a matrix such that AA = A), then the trace of this matrix gives us the number of dimensions it projects onto. Indeed, requiring this property and linearity uniquely* defines trace. So, trace is the linear-algebraization of dimension counting.