Help please with these exercise
Let be a finite-dimensional vector space and let be a linear operator on .
Suppose that rank = rank ,
Prove that the range and null space of are disjoint therefore have only the zero vector in common.
THANKS
Help please with these exercise
Let be a finite-dimensional vector space and let be a linear operator on .
Suppose that rank = rank ,
Prove that the range and null space of are disjoint therefore have only the zero vector in common.
THANKS
1. Prove null-space of = null space of , under the given conditions. (This can be done by showing they have same dimension).
2. Using 1 above establish that if AND implies . As there is a such that . What can you say about and hence and hence from this? That should provide you the prove.