EQUIVALENT STATEMENTS

If A is an n x n matrix, and if is multiplication by A, then the following are equivalent.

1.       A is invertible.

2. has only the trivial solution.

3.       The reduced row-echelon form of A is 4.       A is expressible as a product of elementary matrices.

5. is consistent for every n x 1 matrix 6. has exactly one solution for every n x 1 matrix 7. 8.       The range of 9. is 1 – 1.

10.     The column vectors of A are LI.

11.     The row vectors of A are LI.

12.     The column vectors of A span 13.     The row vectors of A span 14.     The column vectors of A form a basis for 15.     The row vectors of A form a basis for 16.     A has rank n.

17.     A has nullity 0.

18.     The orthogonal complement of the nullspace of A is 19.     The orthogonal complement of the row space of A is 20. is invertible.

21. is not an eigenvalue of A.

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If is a linear system of m equations in n unknowns, then the following are equivalent.

1. is consistent for every m x 1 matrix 2.       The column vectors of A span 3. *************************************************************

If A is an m x n matrix, then the following are equivalent.

1. has only the trivial solution.

2.       The column vectors of A are LI.

3. has at most one solution for every m x 1 matrix 4. is invertible.

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