Linear Algebra

30 Chapter One. Linear Systemš 3.16 For the system2x − y − w = 3y + z + 2w = 2x − 2y − z = −1which of these can be used as the particular solution part of some general solution?⎛ ⎞ ⎛ ⎞ ⎛ ⎞02−1(a) ⎜−3⎟⎝ 5⎠(b) ⎜1⎟⎝1⎠(c) ⎜−4⎟⎝ 8⎠00−1̌ 3.17 Lemma 3.7 says that we can use any particular solution for ⃗p. Find, if possible,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(a) ⎜0⎟⎝0⎠(b) ⎜ 1⎟⎝−7⎠(c) ⎜−1⎟⎝ 1⎠41013.18 One(is nonsingular)while(the)other is singular. Which is which?1 3 1 3(a)(b)4 −12 4 12̌ 3.19 Singular( )or nonsingular?( )1 21 2(a)(b)(c)1 3 −3 −6⎛ ⎞ ⎛ ⎞1 2 12 2 1(d) ⎝1 1 3⎠(e) ⎝ 1 0 5⎠3 4 7−1 1 4( ) 1 2 1(Careful!)1 3 1̌ 3.20 Is(the given(vector(in the set generated by the given set?2 1 1(a) , { , }3)4)5)⎛ ⎞ ⎛ ⎞ ⎛ ⎞−1 2 1(b) ⎝ 0⎠ , { ⎝1⎠ , ⎝0⎠}1 0 1⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 1 2 3 4(c) ⎝3⎠ , { ⎝0⎠ , ⎝1⎠ , ⎝3⎠ , ⎝2⎠}0 4 5 0 1⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 2 3(d) ⎜0⎟⎝ ⎠ , { ⎜1⎟⎝ ⎠ , ⎜0⎟⎝ ⎠ }1101023.21 Prove that any linear system with a nonsingular matrix of coefficients has asolution, and that the solution is unique.3.22 In the proof of Lemma 3.6, what happens if there are no non-‘0 = 0’ equations?̌ 3.23 Prove that if ⃗s and ⃗t satisfy a homogeneous system then so do these vectors.