I am self-studying Linear Algebra from Hoffman and Kunze. The Exercise $6$ on page $5$ asks to show that if two homogeneous systems of linear equations in two unknows have the same solutions, then they are equivalent.

Let $\left\{\begin{array}{c} A_{11}x + A_{12}y = 0 \\ A_{21}x + A_{22}y = 0 \\ \vdots\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \vdots\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \vdots\\ A_{n1}x + A_{n2}y = 0 \end{array}\right.$ and $\left\{\begin{array}{c} B_{11}x + B_{12}y = 0 \\ B_{21}x + B_{22}y = 0 \\ \vdots\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \vdots \,\,\,\,\,\,\,\,\,\,\vdots\\ B_{m1}x + B_{m2}y = 0 \end{array}\right.$ be the two homogeneous systems of linear equations in two unknows. I wrote the fisrt equation of first system as linear combination of the equations of the second system and I get the following:

$\left\{\begin{array}{c} c_{1}B_{11}+c_{2}B_{21}+\cdots+c_{m}B_{m1} = A_{11} \\ c_{1}B_{12}+c_{2}B_{22}+\cdots+c_{m}B_{m2} = A_{12} \end{array}\right.$

I know that I am supposed to find $c_{1},\dots,c_{m}$, but all I know so far is the following: The definition of system of linear equations, solution of the system, homogeneous system of linear equations, and equivalent system and the definition of equivalent system:

Two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations in the other system.

After bgins and Parsa's hints, I was able to solve this question.

PS. The same question appears here, but it was solved by using more than I have so far in the book.


marked as duplicate by ahulpke, Leucippus, user99914, hardmath, Claude Leibovici Jan 15 '18 at 9:25

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  • $\begingroup$ $A$ and $B$ are $n\times2$ matrices. Let $A_i,~B_i$ be their columns for $i=1,2$. Then $A_1x+A_2y=0 \iff B_1x+B_2y=0$. Hmmm. Page 5? So you don't yet know about rank and nullity? Consider two cases: the only solution is $(0,0)$, and there are solutions with $x$ or $y$ nonzero. $\endgroup$ – bgins Feb 18 '12 at 15:24
  • $\begingroup$ @bgins: It is just the beginning of the book! I will work on your suggestion. $\endgroup$ – spohreis Feb 18 '12 at 15:44
  • $\begingroup$ How many independent equations can you have in a system with two variables? Either there are two, in which case all the others can be reduced to $0x+0y=0$ and are irrelevant, or there is one, in which case either $x=0$ (i.e. the columns $A_2=B_2=0$), $y=0$ (i.e. $A_1=B_1=0$), or or $y=ax$ (in which case $A_1=-A_2$ and $B_1=-B_2$), or else there are none, so that $A=B=0$ and all $(x,y)$ are solutions. $\endgroup$ – bgins Feb 18 '12 at 16:01
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    $\begingroup$ Why don't you induct on $n$? $\endgroup$ – Parsa Feb 18 '12 at 20:55
  • $\begingroup$ @Parsa: It worked out! $\endgroup$ – spohreis Feb 19 '12 at 14:19

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