Page 298 - 35Linear Algebra
P. 298
298 Kernel, Range, Nullity, Rank
n
1. If v is any vector in R , then the system Mx = v has exactly one
solution.
2. The matrix M is row-equivalent to the identity matrix.
3. If v is any vector in V , then L(x) = v has exactly one solution.
4. The matrix M is invertible.
5. The homogeneous system Mx = 0 has no non-zero solutions.
6. The determinant of M is not equal to 0.
7. The transpose matrix M T is invertible.
8. The matrix M does not have 0 as an eigenvalue.
9. The linear transformation L does not have 0 as an eigenvalue.
10. The characteristic polynomial det(λI − M) does not have 0 as a root.
n
11. The columns (or rows) of M span R .
12. The columns (or rows) of M are linearly independent.
n
13. The columns (or rows) of M are a basis for R .
14. The linear transformation L is injective.
15. The linear transformation L is surjective.
16. The linear transformation L is bijective.
Note: it is important that M be an n × n matrix! If M is not square,
then it can’t be invertible, and many of the statements above are no longer
equivalent to each other.
Proof. Many of these equivalences were proved earlier in other chapters.
Some were left as review questions or sample final questions. The rest are
left as exercises for the reader.
Invertibility Conditions
298
