Span

The Span of a [set] of vectors is the set: vector,linear combination application example:

The typical well at a cocktail bar contains at least four ingre- dients at the bartender’s disposal: vodka, tequila, orange juice, and grenadine. Assuming we have this well, we can represent drinks as points in , with one element for each in- gredient. For instance, a tequila sunrise can be represented using the point (0, 1.5, 6, 0.75),

representing amounts of vodka, tequila, orange juice, and grenadine (in ounces), respec- tively.

The set of drinks that can be made with our well is contained in ,

that is, all combinations of the four basic ingredients. A bartender looking to save time, however, might notice that many drinks have the same orange juice-to-grenadine ratio and mix the bottles. The new simplified well may be easier for pouring but can make fundamentally fewer drinks:

.