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?
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?
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.
A linear map is invertible iff it is bijective, and injective iff its kernel is trivial.
A linear map $A : X \to Y$ between finite dimensional vector spaces of the same dimenson is invertible iff $\ker A = \{ 0\}$.