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Student[LinearAlgebra]

 GenerateEquations
 generate equations from the coefficient Matrix

 Calling Sequence GenerateEquations(A, v, B)

Parameters

 A - $mxn$ Matrix v - list; specifies the n unknowns B - (optional) $mx1$ Vector

Description

 • The GenerateEquations(A, v) command, generates a list of linear equations from the coefficient Matrix A, using the variable names given in v.
 If the optional $mx1$ right-hand side Vector B is included in the calling sequence, then the list of linear equations is equivalent to the Matrix equation $A·x=B$, where the ${x}_{i}$ are equal to the corresponding ${v}_{i}$.  In this case, v must specify exactly n unknowns.
 If the optional right-hand side Vector B is omitted in the calling sequence, and v specifies exactly n unknowns, then the right-hand sides of the m linear equations are set to zero.  The list of equations is then equivalent to $A·x=\mathrm{ZeroVector}\left(m\right)$.
 If the optional right-hand side Vector B is omitted in the calling sequence, and v specifies exactly $n-1$ unknowns, then the right-hand sides of the m linear equations are taken from the last column of the Matrix A.  That is, the Matrix A is treated as an augmented matrix. The list of equations is then equivalent to ${A}_{1..-1,1..-2}·x={A}_{1..-1,-1}$.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right):$
 > $V≔⟨⟨1,2,3⟩|⟨4,5,6⟩|⟨7,8,0⟩⟩$
 ${V}{≔}\left[\begin{array}{ccc}{1}& {4}& {7}\\ {2}& {5}& {8}\\ {3}& {6}& {0}\end{array}\right]$ (1)
 > $\mathrm{GenerateEquations}\left(V,\left[x,y,z\right],⟨0,-1,3⟩\right)$
 $\left[{x}{+}{4}{}{y}{+}{7}{}{z}{=}{0}{,}{2}{}{x}{+}{5}{}{y}{+}{8}{}{z}{=}{-1}{,}{3}{}{x}{+}{6}{}{y}{=}{3}\right]$ (2)
 > $W≔⟨⟨1,2⟩|⟨3,4⟩⟩$
 ${W}{≔}\left[\begin{array}{cc}{1}& {3}\\ {2}& {4}\end{array}\right]$ (3)
 > $\mathrm{GenerateEquations}\left(⟨W|⟨-3,2⟩⟩,\left[a,b\right]\right)$
 $\left[{a}{+}{3}{}{b}{=}{-3}{,}{2}{}{a}{+}{4}{}{b}{=}{2}\right]$ (4)