Question: Show that the softmax function [ text { softmax }: boldsymbol{z} mapsto frac{exp (boldsymbol{z})}{sum_{k} exp left(z_{k} ight)} ] satisfies the invariance property: [ operatorname{softmax}(boldsymbol{z})=operatorname{softmax}(boldsymbol{z}+c times

Show that the softmax function

\[ \text { softmax }: \boldsymbol{z} \mapsto \frac{\exp (\boldsymbol{z})}{\sum_{k} \exp \left(z_{k}\right)} \]

satisfies the invariance property:

\[ \operatorname{softmax}(\boldsymbol{z})=\operatorname{softmax}(\boldsymbol{z}+c \times \mathbf{1}), \text { for any constant } c \]

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