Injective Linear Transformation

Topics: Algebra - Linear Transformation

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

If $T$ is injective, then:

$$T(x_1)=T(x_2)⟹x_1=x_2$$

$$x_1\ne x_2⟹T(x_1)\ne T(x_2)$$

Equivalences §

The following statements are equivalent:

1. $T$ is injective
2. For all linearly independent $S\subseteq V$ $⟹$ $T(S)$ is also linearly independent in $W$
3. $\ker (T)=\{\overline{0}\}$