Just to clarify these concepts: if a homogeneous system of linear equations with four variables z1, z2, z3, and z4 yields a matrix in reduced row echelon form that defines (as an arbitrary example) the linear equations(adsbygoogle = window.adsbygoogle || []).push({});

z1 = z3 + 0.5z4 = t + 0.5s

z2 = 2z3 - z4 = 2t - s

z3 = t

z4 = s

then the linear combination should be:

[tex]

\left[ \begin{array}{ c } z1 & z2 & z3 & z4 \end{array} \right] = t \left[ \begin{array}{ c } 1 & 2 & 1 & 0 \end{array} \right] + s \left[ \begin{array}{ c } 0.5 & -1 & 0 & 1 \end{array} \right]

[/tex]

and the basic solutions are:

[tex]

\left[ \begin{array}{ c } 1 & 2 & 1 & 0 \end{array} \right] , \left[ \begin{array}{ c } 0.5 & -1 & 0 & 1 \end{array} \right]

[/tex]

My main problem is understanding exactly what the basic solutions are. I'm not sure whether they're just the direction vectors multiplied by each of the parameters or whether they include the parameters as well. Please clarify this for me.

Thanks.

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# Basic Solutions and Linear Combinations

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