Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.  *Two systems of linear equations are $\textit{equivalent}$ if each equation in each system is a linear combination of the equations in the other system.  $\textbf{Proof}.$ Let $$a_{11}x+ a_{12}y=0$$ $$a_{21}x+ a_{22}y=0$$ be a homogeneous system of linear equations where $[x,y]$ is a solution. In addition, let $$b_{11}x+ b_{12}y=0$$ $$b_{21}x+ b_{22}y=0$$ be a homogeneous system of linear equations which has the same solution as the above system.  To show that they are equivalent we have to show that each equation in each system is a linear combination of the equations on the other system. For example,  $$a_{11}x+ a_{12}y= c_1 (b_{11}x+ b_{12}y) + c_2 (b_{21}x+ b_{22}y)$$  where $c_1$ and $c_2$ are weights.  The crucial part here is that the two systems have the same solution so that it is possible to write,  $$(c_1 b_{11} + c_2 b_{21} - a_{11})x + (c_1b_{12} + c_2b_{22} - a_{12})y = 0$$.  Now, if $[x,y] \neq [0,0]$ then it would easily follow that each component $a_{ij}$ is a linear combination of the components $b_{ij}$. But, if $[x,y] = [0,0]$ then e.g. $(c_1 b_{11} + c_2 b_{21} - a_{11}) \neq 0$. How should I proceed to prove this part?
I've just begun to work learn Linear Algebra on my own through Hoffman and Kunze's book and the first problem set already has a question that I can't solve:  Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.  I can't seem to figure out how to prove this without resorting to case work where you account for the cases where one of the coefficients are zero and when both are.  Is there an elegant way to prove in general that when two systems of linear equations have the same solutions, they are equivalent? The converse is obvious enough though.  Definition of equivalence from the text :     Let us say that two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations   in the other system.
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How typeset a numeric table from a CSV file I have a CSV file (db.txt)like this:  30.0,   0.0,    0.0 60.0,   1.9098, 5.8779 90.0,   6.9098, 9.5106 120.0,  13.09,  9.5106 150.0,  18.09,  5.8779 180.0,  20.0,   0.0   I need to typeset it. My problem is related with the wrong alignment of decimal point and the number of decimals (must be a constant for each column):  \renewcommand*{\dtlrealalign}{r}  \DTLloaddb[     noheader,     keys={x,y,theta},     headers={         \shortstack{$\boldsymbol{\theta_{2,i}}$},         \shortstack{X},         \shortstack{Y}              }   ]{db}{db.txt} \begin{table}[t]   \caption{Database file}   \centering   \DTLdisplaydb{db} \end{table}     Any advice, please?