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I was compiling a list of non-singular equivalencies. This is all I have so far. I would appreciate if you can help me to add more to these.
Let A be a square n x n matrix. The following statements are equivalent. That is, for a given A, the statements are either all true or all false.
1. A is non-singular.
2. A is row equivalent to In.
3. Ax=0 has only the trivial solution.
4. Ax=b has a unique solution, for each vector b in Rn .
5. Ax=b has at least one solution, for each vector b in Rn .
6. det(A) ≠ 0
7. The column vectors of A form a linearly independent set in Rn.
8. A(transpose) is non-singular.
9. The column vectors of A span Rn.
10. The column vectors of A form a basis for Rn.
I am thinking in the range of topics like basis, dimension, rank, column space, row space, null space, etc.
any help is highly appreciated.
Let A be a square n x n matrix. The following statements are equivalent. That is, for a given A, the statements are either all true or all false.
1. A is non-singular.
2. A is row equivalent to In.
3. Ax=0 has only the trivial solution.
4. Ax=b has a unique solution, for each vector b in Rn .
5. Ax=b has at least one solution, for each vector b in Rn .
6. det(A) ≠ 0
7. The column vectors of A form a linearly independent set in Rn.
8. A(transpose) is non-singular.
9. The column vectors of A span Rn.
10. The column vectors of A form a basis for Rn.
I am thinking in the range of topics like basis, dimension, rank, column space, row space, null space, etc.
any help is highly appreciated.