Please help Consider the following systems of m equations in n unknowns: Answer

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Consider the following systems of m equations in n unknowns: Answer each of the following questions with sufficient explanation. (a) Suppose n = 29. m = 27. Suppose also that (x1 ... xn) = (1 ... 5) and (x1 ... xn) (0 .. - 4) are two linearly independent solutions of (I) and that all other solutions can be written as a linear combination of these two. Can we choose b1. ... bm in a way that (II) has no solutions? In a way that (II) has a unique solution? Same questions as above with the sole difference that n = 28 instead of 29. Suppose it = 81, m = 76 and that (II) has a solution for any choice of b1..., bm. How many linearly independent solutions of (I) are there? (Bonus) Suppose n = 31, m = 30. Suppose also that (x1 ... xn) = (1 ... 5) and (x1 ... xn) (0 .. - 4) are two linearly independent solutions of (I) and that all other solutions can be written as a linear combination of these two. Can I remove one of the equations in (II) so that the resulting system has a solution for any choice of the right hand sides (i.e. for any choice of the remaining 29 bis)?

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