What is a vector space?

a vector space is a set that's closed under a finite/determined vector addition and scalar multiplication. Basically, scalars are maembers (they can be numbers or variables for exg) in a field called (for exg) F, where in that case, a vector space called (for exg) S would be over (so larger & including) the field we called F.

Answered by alialioxenfree | 2024-06-10

A vector space is a mathematical structure that consists of a set of vectors and two operations: vector addition and scalar multiplication. The set of vectors must satisfy certain properties, such as closure under addition and scalar multiplication. In other words, if you add two vectors together or multiply a vector by a scalar (a real number), the result should still be a vector in the set.
For example, the set of all 2-dimensional vectors with real number entries, denoted as R², is a vector space. In R², you can add two vectors by adding their corresponding entries and scalar multiply a vector by multiplying each entry by a real number.
Some key properties of vector spaces include the existence of a zero vector (the vector that when added to any vector yields the same vector) and the existence of additive inverses (for every vector, there exists a vector that when added to it gives the zero vector).

Answered by qwarrow | 2024-06-24

A vector space is a mathematical structure defined as a set of vectors that can be added and scaled by numbers (scalars) while satisfying specific properties. Key properties include closure under addition and scalar multiplication, existence of a zero vector and inverses, and various associative and distributive laws. An example of a vector space is R n , the set of all n-tuples of real numbers.
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Answered by alialioxenfree | 2025-02-11