In this thesis, the problem of discrete observability of the unforced heat equation is studied. It is explicitly shown that under certain conditions the observability of the heat equation is preserved by two spatial samples and an infinite set of discrete temporal samples. It is shown that the special matrix A = (aij)ooxoo where aij = exp{—i^tj) , tj^s are real and tj < tj+i for j = 1,2,3,...,I= 0,1,2, is a one to one linear operator on the space £°°. In the special case that tj = j \ the explicit form of the inverse of the principal nxn submatrix of the matrix A is calculated for all finite n. This provides a setting in which discrete observability can be verified by explicit estimates.