Vector space

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Vector addeetion an scalar multiplication: a vector v (blue) is addit tae anither vector w (reid, upper illustration). Ablo, w is stretched bi a factor o 2, yieldin the sum v + 2·w.

A vector space is a mathematical structure furmed bi a collection o elements cried vectors, which mey be addit thegither an multiplied ("scaled") bi nummers, cried scalars in this context. Scalars are eften taken tae be real nummers, but thare are an aa vector spaces wi scalar multiplication bi complex nummers, rational nummers, or generally ony field. The operations o vector addeetion an scalar multiplication must satisfy certain requirements, cried axioms. An example o a vector space is that o Euclidean vectors, which mey be uised tae represent pheesical quantities such as forces: ony twa forces (o the same type) can be addit tae yield a third, an the multiplication o a force vector bi a real multiplier is anither force vector. In the same vein, but in a mair geometric sense, vectors representin displacements in the plane or in three-dimensional space an aa furm vector spaces. Vectors in vector spaces dae nae necessarily hae tae be arrow-lik objects as thay appear in the mentioned examples: vectors are best thoucht o as abstract mathematical objects wi particular properties, which in some cases can be visualized as arrows.