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Math 371 Problem Set

Diagonalization/Rank of a matrix/Linear systems

1) For find elementary transformation
such that has the canonical form given in the class with D a diagonal matrix and
corresponding zero matrices, in the following cases:

2) Let and
and considerProve the following:

a) Suppose that for some one has det Then for all
there exist such that det

b) rank(A) is the maximum over all the k such that det for some

3) Analyze which of the following matrices are diagonalizable, where the coefficients are in an arbitrary field,
and if so, find the diagonal form:

4) Discuss the following systems of linear equations, where the coefficients are from an arbitrary commutative
ring R with 1R:

Algebraic field extensions

• Problems 4, 5, 6, 8, 9, at the end of Section 5.1, Ch.5, of Herstein’s book Topics in Algebra.