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30 Chapter One. <strong>Linear</strong> Systemswhich of these can be used as the particular solution part of some general solution?⎛ ⎞0⎛ ⎞2⎛ ⎞−1⎜−3⎟⎜1⎟⎜−4⎟(a) ⎝5⎠ (b) ⎝1⎠ (c) ⎝8⎠00−1̌ 3.18 Lemma 3.8 says that any particular solution may be used for ⃗p. Find, ifpossible, a general solution to this systemx − y + w = 42x + 3y − z = 0y + z + w = 4that uses⎛the⎞given vector⎛ ⎞as its particular⎛ ⎞solution.0−52⎜0⎟⎜ 1 ⎟ ⎜−1⎟(a) ⎝0⎠ (b) ⎝−7⎠ (c) ⎝1⎠41013.19 One of these is nonsingular while the other is singular. Which is which?( ) ( )1 31 3(a)(b)4 −124 12̌ 3.20 Singular ( ) or nonsingular? ( )1 21 2(a)(b)1 3−3 −6( ) 1 2 1(d) 1 1 3 (e)3 4 7( 2 2 11 0 5−1 1 4(c))( )1 2 1(Careful!)1 3 1̌ 3.21 Is( the given ( vector ( in the set generated by the given set?2 1 13)4)5)(a)(b), {( ) −101)2, {(10),)( 1, 01)})}))( ( ( ( 1 1 2 3 4(c) 3 , {(0 , 1 , 3 , 2 }0 4 5 0 1⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 2 3⎜0⎟⎜1⎟⎜0⎟(d) ⎝1⎠ , { ⎝0⎠ , ⎝0⎠}1 1 23.22 Prove that any linear system with a nonsingular matrix of coefficients has asolution, 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 trickypoints after this one.)̌ 3.24 Prove that if ⃗s and ⃗t satisfy a homogeneous system then so do these vectors.(a) ⃗s + ⃗t (b) 3⃗s (c) k⃗s + m⃗t for k, m ∈ RWhat’s wrong with: “These three show that if a homogeneous system has onesolution then it has many solutions — any multiple of a solution is another solution,
Section I. Solving <strong>Linear</strong> Systems 31and any sum of solutions is a solution also — so there are no homogeneous systemswith exactly one solution.”?3.25 Prove that if a system with only rational coefficients and constants has asolution then it has at least one all-rational solution. Must it have infinitely many?
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