I am having some difficulty seeing why the following statement is true:

If $\mathrm{ker}{\chi}_{1}=\mathrm{ker}{\chi}_{2}$, then since ${\chi}_{1}(\mathbf{\text{1}})={\chi}_{2}(\mathbf{\text{1}})=\mathbf{\text{1}}$, we have that ${\chi}_{1}={\chi}_{2}$.