Question #13065

Suppose a nonhomogenous system of 9 linear equations and 10 variables has a solution for all possible values. Discuss whether it is possible for two nonzero solutions that are not multiples of each other to exist for the associated homogenous system.

Expert's answer

As system has solution for all possible value, then it has solution for zeros in

right part, i.e. for homogenous system.

Solutions of associated system will

be dependent from one fixed variable. Thus, value of this variable

determine

the whole solution. So, all solutions will be linearly dependent,

and answer on your question is negative.

right part, i.e. for homogenous system.

Solutions of associated system will

be dependent from one fixed variable. Thus, value of this variable

determine

the whole solution. So, all solutions will be linearly dependent,

and answer on your question is negative.

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