Is it possible to find matrix A satisfying certain conditions?

[songoku]

Homework Statement

Is it possible to find A being a m by n matrix, and two vectors b and c, such that Ax = b has no solution and $A^T$ y = c has exactly one solution? Explain why.

Relevant Equations

Maybe Rank

Since Ax = b has no solution, this means rank (A) < m.

Since $A^T y=c$ has exactly one solution, this means rank ($A^T$) = m

Since rank (A) $\neq$ rank ($A^T$) so matrix A can not exist. Is this valid reasoning?

Thanks

 

[fresh_42]

Homework Statement: Is it possible to find A being a m by n matrix, and two vectors b and c, such that Ax = b has no solution and $A^T$ y = c has exactly one solution? Explain why.
Relevant Equations: Maybe Rank

Since Ax = b has no solution, this means rank (A) < m.

Since $A^T y=c$ has exactly one solution, this means rank ($A^T$) = m

Since rank (A) $\neq$ rank ($A^T$) so matrix A can not exist. Is this valid reasoning?

Thanks

Looks ok to me.

 

[songoku]

Thank you very much fresh_42