# Solving several linear equations

The title might not be too descriptive; I'm not sure how to classify this question.

I have a set of linear equations of the form $$c_iA_i + p_iB_i = E$$ where $A_i$, $B_i$ and $E$ are 3 by 1 column vectors and $c_i$ and $p_i$ are scalars. As the subscripts indicate, I have $n$ such linear equations, where all the scalars and vectors except $E$ change with each one. Another limitation is that all I know for each set of linear equations is $A_i$ and $B_i$, and I want to find $E$.

Is there a method for solving these equations?

EDIT The equation above is a simplification of the equation $$R^T_pT_D = c_iA_i - p_iR_DB_i$$ where $R^T_p$ is the inverse of a rotation matrix, $T_D$ is a translation vector and $R_D$ is another rotation matrix. I am interested in finding the rotation matrix $R_p$. Again, this equation describes the relationship between two homogenious points ($A_i$ and $B_i$) in 3D space, where the z component is 1 (i.e., the c and p scalars project the points to somewhere in the 3D space other than (x, y, 1)). I have n point-pairs A and B.

• In your edit, what is "again" referring to? There was no indication in the original question that the $z$-component of $A_i$ and $B_i$ is $1$. – joriki May 11 '11 at 13:41
• the again was meant to indicate that the A and B in the second equation are the same as the A and B in the first equation. Sorry if I didn't phrase myself clearly. – Marius May 12 '11 at 10:12

This is a homogeneous system of $3n$ linear equations for the $2n+3$ variables $c_i$, $p_i$ and the $3$ components of $E$. In general, for $n\ge3$ the trivial solution (all variables $0$) will be the only solution. If $n<3$, or if the equations happen to be linearly dependent and allow non-trivial solutions, you can find them by subtracting $E$ on both sides and writing out the $n$ $3$-component equations as $3n$ equations. Note that since the system is homogeneous, if there are non-trivial solutions, there's a whole vector space of them.