Surjective Linear Transformations

Topics: Algebra - Linear Transformation

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Let $T:V\to W$ be linear.

$T$ is surjective if $\text{im}(T)=W$.

That is, if $T$ is surjective, then $\forall w\in W$ there exists $v\in V$ such that $T(v)=w$.

Equivalences §

The following statements are equivalent:

1. $T$ is surjective
2. $\forall$ set $A$ that spans $V$ ($\text{span}(A)=V$) $⟹\text{span}(T(A))=W$
3. There exists a set $A_0$ that spans $V$ such that $T(A_0)$ spans $W$ (dubious)