A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.

Created almost 9 years ago, updated about 8 hours ago

For $A=[T]_{C\leftarrow B}$, the number of rows equals $\dim W$. Thus $T:V\to W$ is onto iff $\operatorname{rank}A$ equals its row count. This is the full-row-rank criterion. A full-column-rank condition instead characterizes injectivity.

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