If Ker T = 0 then T is not isomorphism

[nautolian]

Homework Statement

Show that if ker T != 0 then T is not an isomorphism.

Homework Equations

The Attempt at a Solution

If Ker T != 0 that means that there are multiple solutions for which T=0 meaning it is not injective and hence not isomorphic? Is that correct? I don't think it is, or is there a better way of proving this? Thanks.

 

[Dick]

Homework Statement

Show that if ker T != 0 then T is not an isomorphism.

Homework Equations

The Attempt at a Solution

If Ker T != 0 that means that there are multiple solutions for which T=0 meaning it is not injective and hence not isomorphic? Is that correct? I don't think it is, or is there a better way of proving this? Thanks.

Well, it is correct. If x is a nonzero element of ker T, then Tx=0=T0. Not injective. Not isomorphism.