Linear Algebra Logical Equivalency

[Kittygodofpower]

Assuming the linear transformation x --> Ax maps R^n onto R^n how can I prove that there is an nxn matrix C such that CA = I? (where I is the nxn identity matrix)

 

[daon2]

Assuming the linear transformation x --> Ax maps R^n onto R^n how can I prove that there is an nxn matrix C such that CA = I? (where I is the nxn identity matrix)

One way is to prove that there exists a matrix C such that A^TC = I. This is true if and only if C^TA=I.

Hint: there exists a vector v_i such that A^T(v_i) = e_i.