Solution Sets

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Parametric Vector Form (homogeneous case)

Consider the following matrix in reduced row echelon form:

A = C 10 − 8 − 701430000 D .

The matrix equation Ax = 0 corresponds to the system of equations

T x 1 − 8 x 3 − 7 x 4 = 0 x 2 + 4 x 3 + 3 x 4 = 0.

We can write the parametric form as follows:

GMKMI x 1 = 8 x 3 + 7 x 4 x 2 = − 4 x 3 − 3 x 4 x 3 = x 3 x 4 = x 4 .

We wrote the redundant equations x 3 = x 3 and x 4 = x 4 in order to turn the above system into a vector equation:

x = EPN x 1 x 2 x 3 x 4 FQO = x 3 EPN 8 − 410 FQO + x 4 EPN 7 − 301 FQO .

This vector equation is called the parametric vector form of the solution set. Since x 3 and x 4 are allowed to be anything, this says that the solution set is the set of all linear combinations of EPN 8 − 410 FQO and EPN 7 − 301 FQO . In other words, the solution set is

Span GMKMIEPN 8 − 410 FQO , EPN 7 − 301 FQOHMLMJ .

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