When you multiply a linear floating point image value by 2, you will get a new value that is twice as bright. Or you can say that .5 is half as bright as 1.
The values have to be stored in floating point. Storing fixed-point (or multiplying by a constant and storing integers) is not an efficient use of memory for storing images, as described here.
High Dynamic Range: Many people think the purpose of floating point is to store High Dynamic Range (HDR) images, meaning images with values outside the 0 to 1 range. It is true that most methods of efficiently storing a floating point value (such as exponent + mantissa) also will store values outside the 0 to 1 range. And my algorithims are able to convert to and from HDR if you use functions other than sRGB. But in fact there are many advantages to linear even if you do not go out of range.
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