Let R + be the set of all positive real numbers excluding zero, and let us defin

Question

Let R+ be the set of all positive real numbers excluding zero, and let us define the operations of addition (+) and scalar multiplication (*) as follows:

Addition: For x,y R+ ,where x and y are vectors, then x (+) y = x y , where x y is the typical product of x and y that we are used to.

Scalar Multiplication: For x R+ and cR (note, all scalars can come from all R), then c (*) x = x^c

where x^c is exponentiation in the way we are used to. Verify that R+ with these operations of

(+) and (*) are a vector space.

Explanation / Answer

let v1, v2 be two vectors in the vector space. then v1(+)v2 = v1v2 also belongs to the vector space. v1(*)c = v1^c = v1*v1*v1*v1...(c times), where c belongs to R hence it also belongs to the vector space. hence the vector space is closed under addition and scalar multiplication...

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