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LetAbe annnmatrix. Prove thatAis nonsingular if and only if the columns ofAspanF^n. You may use the most recent version of the nonsingularity theorem in your
LetAbe annnmatrix. Prove thatAis nonsingular if and only if the columns ofAspanF^n. You may use the most recent version of the nonsingularity theorem in your proof.
Nonsingularity theorem:
LetAbe annnmatrix. The following are equivalent.
- Ais nonsingular.
- ThenullspaceofAisN(A)={0}.
- The row reduced echelon form ofAisIn.
- TherankofAisn.
- The homogeneous system of linear equations with coefficient matrixAhas only the trivial solution.
- The system of linear equations with coefficient matrixAand constant vectorbhas a unique solution for any choice ofb.
- Ais invertible.
- The columns ofAare linearly independent.
- The columns ofAform a basis ofFn.
This is all the information I've been given!!!!
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