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.