Linear Algebra Exercises-n-Answers.pdf
Linear Algebra Exercises-n-Answers.pdf Linear Algebra Exercises-n-Answers.pdf
14 Linear Algebra, by Hefferon One.I.2.31 This is how the answer was given in the cited source. Let u, v, x, y, z be the volumes in cm 3 of Al, Cu, Pb, Ag, and Au, respectively, contained in the sphere, which we assume to be not hollow. Since the loss of weight in water (specific gravity 1.00) is 1000 grams, the volume of the sphere is 1000 cm 3 . Then the data, some of which is superfluous, though consistent, leads to only 2 independent equations, one relating volumes and the other, weights. u + v + x + y + z = 1000 2.7u + 8.9v + 11.3x + 10.5y + 19.3z = 7558 Clearly the sphere must contain some aluminum to bring its mean specific gravity below the specific gravities of all the other metals. There is no unique result to this part of the problem, for the amounts of three metals may be chosen arbitrarily, provided that the choices will not result in negative amounts of any metal. If the ball contains only aluminum and gold, there are 294.5 cm 3 of gold and 705.5 cm 3 of aluminum. Another possibility is 124.7 cm 3 each of Cu, Au, Pb, and Ag and 501.2 cm 3 of Al. Subsection One.I.3: General = Particular + Homogeneous One.I.3.15 For the arithmetic to these, see the answers from the prior subsection. (a) The solution set is ( ( ) 6 −2 { + y 0) ∣ y ∈ R}. 1 Here the particular solution and the ( solution set ( for ) the associated homogeneous system are 6 −2 and { y 0) ∣ y ∈ R}. 1 (b) The solution set is ( 0 { }. 1) The particular solution and the solution ( set for the associated ( homogeneous system are 0 0 and { } 1) 0) (c) The solution set is ⎛ { ⎝ 4 ⎞ ⎛ −1⎠ + ⎝ −1 ⎞ ∣ 1 ⎠ x 3 x3 ∈ R}. 0 1 A particular solution and the solution ⎛ set for the associated homogeneous system are ⎝ 4 ⎞ ⎛ −1⎠ and { ⎝ −1 ⎞ ∣ 1 ⎠ x 3 x 3 ∈ R}. 0 1 (d) The solution set is a singleton ⎛ ⎞ { ⎝ 1 1⎠}. 1 A particular solution and the solution ⎛ ⎞ set for the ⎛ associated ⎞ homogeneous system are ⎝ 1 1⎠ and { ⎝ 0 0⎠ t ∣ t ∈ R}. 1 0 (e) The solution set is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 5/3 −1/3 −2/3 { ⎜2/3 ⎟ ⎝ 0 ⎠ + ⎜ 2/3 ⎟ ⎝ 1 ⎠ z + ⎜ 1/3 ⎟ ⎝ 0 ⎠ w ∣ z, w ∈ R}. 0 0 1 A particular solution and ⎛ the ⎞ solution set ⎛ for the ⎞ associated ⎛ ⎞ homogeneous system are 5/2 −1/3 −2/3 ⎜2/3 ⎟ ⎝ 0 ⎠ and { ⎜ 2/3 ⎟ ⎝ 1 ⎠ z + ⎜ 1/3 ⎟ ⎝ 0 ⎠ w ∣ z, w ∈ R}. 0 0 1
Answers to Exercises 15 (f) This system’s solution set is empty. Thus, there is no particular solution. The solution set of the associated homogeneous system is ⎛ ⎞ ⎛ ⎞ −1 −1 { ⎜ 2 ⎟ ⎝ 1 ⎠ z + ⎜ 3 ⎟ ⎝ 0 ⎠ w ∣ z, w ∈ R}. 0 1 One.I.3.16 The answers from the prior subsection show the row operations. (a) The solution set is ⎛ { ⎝ 2/3 ⎞ ⎛ −1/3⎠ + ⎝ 1/6 ⎞ 2/3⎠ z ∣ z ∈ R}. 0 1 A particular solution and the solution set for the associated homogeneous system are ⎛ ⎝ 2/3 ⎞ ⎛ ⎞ 1/6 −1/3⎠ and { ⎝2/3⎠ z ∣ z ∈ R}. 0 1 (b) The solution set is ⎛ ⎞ ⎛ ⎞ 1 1 { ⎜3 ⎟ ⎝0⎠ + ⎜−2 ⎟ ⎝ 1 ⎠ z ∣ z ∈ R}. 0 0 A particular solution and the solution set for the associated homogeneous system are ⎛ ⎞ ⎛ ⎞ 1 1 ⎜3 ⎟ ⎝0⎠ and { ⎜−2 ⎟ ⎝ 1 ⎠ z ∣ z ∈ R}. 0 0 (c) The solution set is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 −1 −1 { ⎜0 ⎟ ⎝0⎠ + ⎜ 0 ⎟ ⎝ 1 ⎠ z + ⎜−1 ⎟ ⎝ 0 ⎠ w ∣ z, w ∈ R}. 0 0 1 A particular solution and the solution set for the associated homogeneous system are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 −1 −1 ⎜0 ⎟ ⎝0⎠ and { ⎜ 0 ⎟ ⎝ 1 ⎠ z + ⎜−1 ⎟ ⎝ 0 ⎠ w ∣ z, w ∈ R}. 0 0 1 (d) The solution set is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 −5/7 −3/7 −1/7 0 { ⎜0 ⎟ ⎝0⎠ + −8/7 ⎜ 1 ⎟ ⎝ 0 ⎠ c + −2/7 ⎜ 0 ⎟ ⎝ 1 ⎠ d + 4/7 ⎜ 0 ⎟ ⎝ 0 ⎠ e ∣ c, d, e ∈ R}. 0 0 0 1 A particular solution and the solution set for the associated homogeneous system are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 −5/7 −3/7 −1/7 0 ⎜0 ⎟ ⎝0⎠ and { −8/7 ⎜ 1 ⎟ ⎝ 0 ⎠ c + −2/7 ⎜ 0 ⎟ ⎝ 1 ⎠ d + 4/7 ⎜ 0 ⎟ ⎝ 0 ⎠ e ∣ c, d, e ∈ R}. 0 0 0 1 One.I.3.17 Just plug them in and see if they satisfy all three equations. (a) No. (b) Yes. (c) Yes. One.I.3.18 Gauss’ method on the associated homogeneous system gives ⎛ ⎝ 1 −1 0 1 0 ⎞ ⎛ 2 3 −1 0 0⎠ −2ρ 1+ρ 2 −→ ⎝ 1 −1 0 1 0 ⎞ ⎛ 0 5 −1 −2 0⎠ −(1/5)ρ 2+ρ 3 −→ ⎝ 1 −1 0 1 0 ⎞ 0 5 −1 −2 0⎠ 0 1 1 1 0 0 1 1 1 0 0 0 6/5 7/5 0
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<strong>Answers</strong> to <strong>Exercises</strong> 15<br />
(f) This system’s solution set is empty. Thus, there is no particular solution. The solution set of the<br />
associated homogeneous system is<br />
⎛ ⎞ ⎛ ⎞<br />
−1 −1<br />
{ ⎜ 2<br />
⎟<br />
⎝ 1 ⎠ z + ⎜ 3<br />
⎟<br />
⎝ 0 ⎠ w ∣ z, w ∈ R}.<br />
0 1<br />
One.I.3.16 The answers from the prior subsection show the row operations.<br />
(a) The solution set is<br />
⎛<br />
{ ⎝ 2/3<br />
⎞ ⎛<br />
−1/3⎠ + ⎝ 1/6<br />
⎞<br />
2/3⎠ z ∣ z ∈ R}.<br />
0 1<br />
A particular solution and the solution set for the associated homogeneous system are<br />
⎛<br />
⎝ 2/3<br />
⎞ ⎛ ⎞<br />
1/6<br />
−1/3⎠ and { ⎝2/3⎠ z ∣ z ∈ R}.<br />
0<br />
1<br />
(b) The solution set is<br />
⎛ ⎞ ⎛ ⎞<br />
1 1<br />
{ ⎜3<br />
⎟<br />
⎝0⎠ + ⎜−2<br />
⎟<br />
⎝ 1 ⎠ z ∣ z ∈ R}.<br />
0 0<br />
A particular solution and the solution set for the associated homogeneous system are<br />
⎛ ⎞ ⎛ ⎞<br />
1<br />
1<br />
⎜3<br />
⎟<br />
⎝0⎠ and { ⎜−2<br />
⎟<br />
⎝ 1 ⎠ z ∣ z ∈ R}.<br />
0<br />
0<br />
(c) The solution set is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
0 −1 −1<br />
{ ⎜0<br />
⎟<br />
⎝0⎠ + ⎜ 0<br />
⎟<br />
⎝ 1 ⎠ z + ⎜−1<br />
⎟<br />
⎝ 0 ⎠ w ∣ z, w ∈ R}.<br />
0 0 1<br />
A particular solution and the solution set for the associated homogeneous system are<br />
⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
0<br />
−1 −1<br />
⎜0<br />
⎟<br />
⎝0⎠ and { ⎜ 0<br />
⎟<br />
⎝ 1 ⎠ z + ⎜−1<br />
⎟<br />
⎝ 0 ⎠ w ∣ z, w ∈ R}.<br />
0<br />
0 1<br />
(d) The solution set is<br />
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
1 −5/7 −3/7 −1/7<br />
0<br />
{<br />
⎜0<br />
⎟<br />
⎝0⎠ + −8/7<br />
⎜ 1<br />
⎟<br />
⎝ 0 ⎠ c + −2/7<br />
⎜ 0<br />
⎟<br />
⎝ 1 ⎠ d + 4/7<br />
⎜ 0<br />
⎟<br />
⎝ 0 ⎠ e ∣ c, d, e ∈ R}.<br />
0 0<br />
0<br />
1<br />
A particular solution and the solution set for the associated homogeneous system are<br />
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
1<br />
−5/7 −3/7 −1/7<br />
0<br />
⎜0<br />
⎟<br />
⎝0⎠ and { −8/7<br />
⎜ 1<br />
⎟<br />
⎝ 0 ⎠ c + −2/7<br />
⎜ 0<br />
⎟<br />
⎝ 1 ⎠ d + 4/7<br />
⎜ 0<br />
⎟<br />
⎝ 0 ⎠ e ∣ c, d, e ∈ R}.<br />
0<br />
0<br />
0<br />
1<br />
One.I.3.17 Just plug them in and see if they satisfy all three equations.<br />
(a) No.<br />
(b) Yes.<br />
(c) Yes.<br />
One.I.3.18 Gauss’ method on the associated homogeneous system gives<br />
⎛<br />
⎝ 1 −1 0 1 0<br />
⎞ ⎛<br />
2 3 −1 0 0⎠ −2ρ 1+ρ 2<br />
−→ ⎝ 1 −1 0 1 0<br />
⎞<br />
⎛<br />
0 5 −1 −2 0⎠ −(1/5)ρ 2+ρ 3<br />
−→ ⎝ 1 −1 0 1 0<br />
⎞<br />
0 5 −1 −2 0⎠<br />
0 1 1 1 0<br />
0 1 1 1 0<br />
0 0 6/5 7/5 0
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