Bijective Linear Transformations

Topics: Algebra - Linear Transformation - Vector Space

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If a linear transformation $T:V\to W$ is injective and surjective, then it is bijective.

If $T$ is bijective, then we call it an isomorphism and it’s invertible.

Equivalences §

The following statements are equivalent:

1. $T$ is bijective
2. For all bases $\beta$ of $V$, $T(\beta$) is a basis of $W$
3. There exists a basis $B_0$ of $V$ such that $T(B_0)$ is a generating set for $W$ (dubious)