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.      

 

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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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