Linear Algebra

Section I. Solving Linear Systems 31
3.22 Prove that any linear system with a nonsingular matrix of coefficients has a
solution, and that the solution is unique.
3.23 To tell the whole truth, there is another tricky point to the proof of Lemma 3.7.
What happens if there are no non-‘0 = 0’ equations? (There aren’t any more tricky
points after this one.)
� 3.24 Prove that if �s and �t satisfy a homogeneous system then so do these vectors.
(a) �s + �t (b) 3�s (c) k�s + m�t for k, m ∈ R
What’s wrong with: “These three show that if a homogeneous system has one
solution then it has many solutions — any multiple of a solution is another solution,
and any sum of solutions is a solution also — so there are no homogeneous systems
with exactly one solution.”?
3.25 Prove that if a system with only rational coefficients and constants has a
solution then it has at least one all-rational solution. Must it have infinitely many?