Before talking about vector spaces, let’s go back to basic principles. To keep things concrete, we’ll talk physics. One fundamental class of physical properties and phenomena can be described by a single value; for example, mass, volume, distance, or temperature. However, a second class of properties or phenomena cannot be so described; rather, they are by nature multiple-valued. One example of this class is motion of an object, which has two components—speed and direction (as, say, kilometers per hour and compass heading). The first class of properties can be called scalar-valued, and the second class of properties can be called vector-valued. Vector-valued entities may also be referred to as multidimensional ...

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