Linear algebra homework question, please help.

[Tyler2606]

This was all my professor gave me to work with....
Also, I am sure the statement is true but not sure how to put it into words or what to refer too.

An engineer discovers that all solutions to a homogeneous system of nine linear equations in eleven variables can be expressed as a linear combination of two linearly independent solutions. Can the engineer be certain that an associated non-homogeneous system with same coefficient matrix has a solution? explain you answer.

 

[kongzi]

Hi

The answer is No.

We have 9 line linear equations that reveal to no be independent. Therefore at least 1 of them, let's say the last one, is a linear combination of the 8 other equations.

Suppose now that we add different constant in each equation. In this case, the last equation which is dependent with the rest of equations, can't be true (consistent with the other equations) except if the added constant to this equation corresponds to the combination of the added constants in the other equations, following the dependence combination that relates the last equation with the other equations.

 

[Tyler2606]

Hi

The answer is No.

We have 9 line linear equations that reveal to no be independent. Therefore at least 1 of them, let's say the last one, is a linear combination of the 8 other equations.

Suppose now that we add different constant in each equation. In this case, the last equation which is dependent with the rest of equations, can't be true (consistent with the other equations) except if the added constant to this equation corresponds to the combination of the added constants in the other equations, following the dependence combination that relates the last equation with the other equations.

Thank you, this helps a ton.