Problem #5

a) Show that the product of A(AT) IS A SYMMETRIC MATRIX

b) Show that any square matrix A can be written as the sum of a symmetric matrix S and a skew-symmetric T

c) Give an example to show that if the product of the two matrices A and B is the zero matrix, it does not imply that A or B has to be the zero matrix

d) Given the linear system Ax=b , state under what conditions the system has:
i)a unique solution ii) multiple solution iii) no solution

e) What are the two key properties that a vector space V should hold?

f) What are the properties for a basis of a vector space V? Is the basis unique?

g) solve the set of linear equations:

x-y=3
2x-3y=k

For which values of the constant k does the set have no solution? many solutions? unique solution?