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I have a vector of numbers (both positive and negative). How can I normalize it so the values of the normalized vector will be between 0 and 1?

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This is called unity-based normalization. If you have a vector $X$, you can obtain a normalized version of it, say $Z$, by doing:

$$Z = \frac{X - \min(X)}{\max(X) - \min(X)}$$

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