suppose [ an ]where the {ai form basis of rn take any vector [ xn ]> with respect to the standard basis (we write subscript to stress thisthen the representation of this vector with respect to is simply - xwhere - is the inverse of athat isthe matrix such that aa- - in where in is the -dimensional identity matrix to see thisnote that - ai gives the -th unit vector representationfor nand recall that each vector in rn is unique linear combination of these basis vectors example (basis representationconsider the matrix - - awith inverse / - / inverse ( the vector [ ]> in the standard basis has representation - [- ]> in the basis consisting of the columns of namely ay the transpose of matrix [ai is the matrix [ ji ]that isthe (ij)-th element of ais the ji)-th element of the trace of square matrix is the sum of its diagonal elements useful result is the following cyclic property transpose trace theorem cyclic property the trace is invariant under cyclic permutationstr(abctr(bcatr(cabproofit suffices to show that tr(deis equal to tr(edfor any matrix [di and matrix [ei the diagonal elements of de are nj= di ji and the diagonal elements of ed are mi= ji di they sum up to the same number pm pn = = di ji square matrix has an inverse if and only if its columns (or rowsare linearly independent this is the same as the matrix being of full rankthat isits rank is equal to the number of columns an equivalent statement is that its determinant is not zero the determinant of an matrix [aij is defined as det( : (- (pn api , determinant ( = where the sum is over all permutations ( pn of ( )and (pis the number of pairs (ijfor which for examplez( for the pairs ( )( )( the determinant of diagonal matrix - matrix with only zero elements off the diagonal -is simply the product of its diagonal elements diagonal matrix