Thread: Advance linear transformation exercise

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

Originally Posted by osodud

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) by the rank-nullity theorem we also know that the null space of is a subspace of the null space of so the null space of is equal to the null space of

now if is in the null space of then and thus by (1).

Originally Posted by osodud

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.