TITLE: Relationship between null space and invertibility of linear transformations
QUESTION [2 upvotes]: 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?

REPLY [0 votes]: A linear map $A : X \to Y$ between finite dimensional vector spaces of the same dimenson is invertible iff $\ker A = \{ 0\}$.
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