Show that $\text{ker}(T^*T)=\text{ker}(T)$

Question

Show that $\text{ker}(T^*T)=\text{ker}(T)$

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In my solutions when showing that $x\in\text{ker}(T)\implies x\in\text{ker}(T^*T)$ the following is given:

$$x\in\text{ker}(T)\implies Tx=0\implies T^*Tx=0\implies x\in\text{ker}(T^*T)$$

How exactly does $Tx=0\implies T^*Tx=0$ ?

Answer 1

$T^*$ is a linear operator, hence $T^*(0)=0$.