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melisa1 [442]
3 years ago
14

Explain how to determine if something is a vector space or not

Mathematics
1 answer:
nlexa [21]3 years ago
6 0
All you do is referring to the following definition:

Definition: A<span> vector space </span>is a set V on which two operations + and · are defined, called<span> vector addition </span>and<span> scalar multiplication.</span>

The operation + (vector addition) must satisfy the following conditions:

Closure: If u and v are any vectors in V, then the sum   u + v   belongs to V.

(1)<span> Commutative law: </span>For all vectors u and v in V,     u + v = v + u

(2)<span> Associative law: </span>For all vectors u, v, w in V,     u <span>+ (v</span> + w<span>) = (u</span> + v) + w

(3)<span> Additive identity: </span>The set V contains an<span> additive identity </span>element, denoted by 0, such that for any vector v in V,     0 + v = v   and   v + 0 = v.

(4)<span> Additive inverses: </span>For each vector v in V, the equations     v + x = 0   and   x + v = 0     have a solution x in V, called an<span> additive inverse </span>of v, and denoted by - v.

The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:

Closure: If v in any vector in V, and c is any real number, then the product   c · v   belongs to V.

(5)<span> Distributive law: </span>For all real numbers c and all vectors u, v in V,     c · <span>(u</span> + v) = c · u + c · v

(6)<span> Distributive law</span>: For all real numbers c, d and all vectors v in V,     (c+d) · v = c · v + d · v

(7)<span> Associative law</span>: For all real numbers c,d and all vectors v in V,     c · (d · v) = (cd) · v

(8)<span> Unitary law</span>: For all vectors v in V,     1 · v = <span>v</span>

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