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What is a vector space?

A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is -dimensional Euclidean space , where every element is represented by a list of real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.

Who invented vector spaces?

Vector spaces as abstract algebraic entities were first defined by the Italian mathematician Giuseppe Peano in 1888. Peano called his vector spaces “linear systems” because he correctly saw that one can obtain any vector in the space from a linear combination of finitely many vectors and scalars— a v + b w + … + c z.

Is v w a vector space?

or in words, all ordered pairs of elements from V and W. In fact V × W is a vector space if V and W are. We have actually been using this fact already:

What is the difference between a covector and a vector?

Covector, an element of the dual of a vector space. In an inner product space, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a covector from a vector. The distinction becomes apparent when one changes coordinates (non-orthogonally).

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