Subsection 19.3

Equivalent systems of equations (19.21-19.29)

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Problem 19.21

Prove that if consistent systems of linear inhomogeneous equations are equivalent, then the corresponding homogeneous systems are also equivalent.

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Problem 19.22
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(1)

Prove that the nontrivial equations1 a1x1+…+anxn=0a_{1} x_{1} + \ldots + a_{n} x_{n} = 0 and b1x1+⋯+bnxn=0b_{1} x_{1} + \cdots + b_{n} x_{n} = 0 are equivalent if and only if rg⁡[a1…anb1…bn]=1\operatorname {rg}\left[\begin{smallmatrix} a_{1} & \ldots & a_{n} \\ b_{1} & \ldots & b_{n}\end{smallmatrix}\right] = 1.

Footnotes

  1. A linear equation is nontrivial if at least one of the coefficients of the unknowns is different from 0. ↩

(2)

Prove that the nontrivial1{ }^{1}) equations a1x1+…⋯+anxn=aa_{1} x_{1} + \ldots \cdots + a_{n} x_{n} = a and b1x1+⋯+bnxn=bb_{1} x_{1} + \cdots + b_{n} x_{n} = b are equivalent if and only if rg⁡[a1…anab1…bnb]=1\operatorname {rg}\left[\begin{smallmatrix} a_{1} & \ldots & a_{n} & a \\ b_{1} & \ldots & b_{n} & b\end{smallmatrix}\right] = 1.

(3)

Formulate a criterion for pairwise equivalence of kk linear equations.

Problem 19.23
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(1)

Prove that the systems of linear equations Ax=o,Bx=oA\mathbf{x} = \mathbf{o}, B\mathbf{x} = \mathbf{o} are equivalent if and only if

rg⁡[AB]□=rg⁡A=rg⁡B. \operatorname {rg}\left[\begin{smallmatrix} A \\ B\end{smallmatrix}\right]^{\square } = \operatorname {rg} A = \operatorname {rg} B.
(2)

Prove that the consistent systems of linear equations Ax=a,Bx=bA\mathbf{x} = \mathbf{a}, B\mathbf{x} = \mathbf{b} are equivalent if and only if

rg⁡[AaB b]□=rg⁡A=rg⁡B. \operatorname {rg}\left[\begin{smallmatrix} A & \mathbf{a} \\ B & \mathrm{~ b} \end{smallmatrix}\right]^{\square } = \operatorname {rg} A = \operatorname {rg} B.
Problem 19.24

Check the equivalence of the systems of equations 18.1,1118. 1, 11) and 18.1,12)18. 1, 12).

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Problem 19.25

Check whether the systems of equations defined by the augmented matrices are equivalent:

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(1)

(A502∣c17)\left(\mathbf{A}_{502} \mid \mathbf{c}_{17}\right) and (A503∣c12)\left(\mathbf{A}_{503} \mid \mathbf{c}_{12}\right). A502=[117−31−1−11],c17=[20],A503=[3−2112−120],c12=[11]\mathbf{A}_{502} = \left[\begin{smallmatrix} 1 & 1 & 7 & -3 \\ 1 & -1 & -1 & 1 \end{smallmatrix}\right], \mathbf{c}_{17} = \left[\begin{smallmatrix} 2 \\ 0 \end{smallmatrix}\right], \mathbf{A}_{503} = \left[\begin{smallmatrix} 3 & -2 & 1 & 1 \\ 2 & -1 & 2 & 0 \end{smallmatrix}\right], \mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right].

(2)

(A239∣c67)\left(\mathbf{A}_{239} \mid \mathbf{c}_{67}\right) and (A240∣c147)\left(\mathbf{A}_{240} \mid \mathbf{c}_{147}\right). A239=[012−10−2−220],c67=[012],A240=[−54−22161−2−2],c147=[5−2−1]\mathbf{A}_{239} = \left[\begin{smallmatrix} 0 & 1 & 2 \\ -1 & 0 & -2 \\ -2 & 2 & 0 \end{smallmatrix}\right], \mathbf{c}_{67} = \left[\begin{smallmatrix} 0 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{A}_{240} = \left[\begin{smallmatrix} -5 & 4 & -2 \\ 2 & 1 & 6 \\ 1 & -2 & -2 \end{smallmatrix}\right], \mathbf{c}_{147} = \left[\begin{smallmatrix} 5 \\ -2 \\ -1 \end{smallmatrix}\right].

(3)

(A581∣c69)\left(\mathbf{A}_{581} \mid \mathbf{c}_{69}\right) and (A422T∣c72)\left(\mathbf{A}_{422}^{T} \mid \mathbf{c}_{72}\right). A581=[135791−23−45211122522],c69=[−26−11−67],A422=[4192−11−15928−20135−342],c72=[−513]\mathbf{A}_{581} = \left[\begin{smallmatrix} 1 & 3 & 5 & 7 & 9 \\ 1 & -2 & 3 & -4 & 5 \\ 2 & 11 & 12 & 25 & 22 \end{smallmatrix}\right], \mathbf{c}_{69} = \left[\begin{smallmatrix} -26 \\ -11 \\ -67 \end{smallmatrix}\right], \mathbf{A}_{422} = \left[\begin{smallmatrix} 4 & 1 & 9 \\ 2 & -11 & -15 \\ 9 & 2 & 8 \\ -20 & 13 & 5 \\ -3 & 4 & 2 \end{smallmatrix}\right], \mathbf{c}_{72} = \left[\begin{smallmatrix} -5 \\ 1 \\ 3 \end{smallmatrix}\right].

Problem 19.26

Check the equivalence of all systems in the given collection (each system of equations is given by an augmented matrix): (A501∣c16),(A509∣c67)\left(\mathbf{A}_{501} \mid \mathbf{c}_{16}\right), \left(\mathbf{A}_{509} \mid \mathbf{c}_{67}\right) and (A510∣c73)\left(\mathbf{A}_{510} \mid \mathbf{c}_{73}\right). A501=[187−151−5−611],c16=[17−9],A509=[2−1−34−325−7011−2],c67=[012],A510=[1−5−61151−43187−15],c73=[−9717]\mathbf{A}_{501} = \left[\begin{smallmatrix} 1 & 8 & 7 & -15 \\ 1 & -5 & -6 & 11 \end{smallmatrix}\right], \mathbf{c}_{16} = \left[\begin{smallmatrix} 17 \\ -9 \end{smallmatrix}\right], \mathbf{A}_{509} = \left[\begin{smallmatrix} 2 & -1 & -3 & 4 \\ -3 & 2 & 5 & -7 \\ 0 & 1 & 1 & -2 \end{smallmatrix}\right], \mathbf{c}_{67} = \left[\begin{smallmatrix} 0 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{A}_{510} = \left[\begin{smallmatrix} 1 & -5 & -6 & 11 \\ 5 & 1 & -4 & 3 \\ 1 & 8 & 7 & -15 \end{smallmatrix}\right], \mathbf{c}_{73} = \left[\begin{smallmatrix} -9 \\ 7 \\ 17 \end{smallmatrix}\right].

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Problem 19.27
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(1)

Suppose that adjoining to a given homogeneous system of linear equations some further number of linear homogeneous equations does not change the set of its solutions. Prove that the adjoined equations are linear combinations of the equations of the given system.

(2)

Prove the same statement for a consistent system of linear inhomogeneous equations. Compare with problem 17. 3 2)

Problem 19.28
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(1)

Suppose that every solution of the homogeneous system of linear equations (A) is also a solution of the homogeneous system of linear equations (B). Prove that then every equation of system (B) is a linear combination of the equations of system (A).

(2)

Prove the same statement for consistent systems of linear inhomogeneous equations. Compare with problem 17.3(1).

Problem 19.29
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(1)

Prove that two homogeneous systems of linear equations are equivalent if and only if the equations of each of them are linear combinations of the equations of the other system.

(2)

Prove the same statement for consistent systems of linear inhomogeneous equations.