Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a relationship between the null space $N(T)$ of a linear transformation $T$ and whether or not it is invertible?

For example, if you know $N(T) \neq \{0\}$, can you be sure it's not an invertible transformation?

share|cite|improve this question
If you know the null space is nonzero, then it certainly isn't invertible, since more than one element maps to zero (so to what element would the inverse map zero?). – Alan Guo Dec 9 '12 at 18:31
Yes that's true - I didn't think of that! – CodyBugstein Dec 9 '12 at 18:35
up vote 4 down vote accepted

Yes, what is true is that $T:V\to W$ is injective if and only if $N(T)=\{0\}$. Thus, if you know that $N(T)\ne\{0\}$ you know that $T$ can't be injective, and so particularly can't be invertible. Note that $N(T)=\{0\}$ is not sufficient for invertibility as the map $T:\mathbb{R}^3\to\mathbb{R}^4:(x,y,z)\mapsto (x,y,z,0)$ shows.

share|cite|improve this answer

A linear map is invertible iff it is bijective, and injective iff its kernel is trivial.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.