Let be the standard basis vectors of . A second set of vectors are defined with respect to the standard basis by
The are the elements of the matrix . State the condition on under which the set forms a basis of .
Define the matrix that, for a given linear transformation , gives the relation between the components of any vector and those of the corresponding , with the components specified with respect to the standard basis.
Show that the relation between the matrix and the matrix of the same transformation with respect to the second basis is
Consider the matrix
Find a matrix such that is diagonal. Give the elements of and demonstrate explicitly that the relation between and holds.
Give the elements of for any positive integer .