Determine if the described set is a subspace. The subset of R2 consisting of vectors of the form a b , where a and b are rational numbers. The set is a subspace. The set is not a subspace. If so, give a proof. If not, explain why not.

Question

contestada

Respuesta :

berno berno · Answer 1 · 2020-02-20T08:21:40+01:00

Answer:

Yes, W is a subspace.

Step-by-step explanation:

Let V be a vector space and W be a subset of V.

W is said to be a subspace of the vector space V if it satisfies the following conditions:

1. W is non-empty.

2. If [tex]x,y\varepsilon W[/tex] then [tex]x+y\varepsilon W[/tex]

3. If [tex]x\varepsilon W[/tex] then [tex]kx\varepsilon W[/tex] where k is a scalar.

Solution:

Let [tex]W=\left \{ ab;\,a\,,b\,\varepsilon Q \right \}[/tex]

Here, Q denotes the set of rational numbers.

W is non-empty as [tex]0\varepsilon W[/tex] being a rational number.

Let [tex]x=ab\,,\,y=cd\,\,\varepsilon W[/tex] where a, b, c, d are rational numbers.

[tex]x+y=ab+cd\,\varepsilon\, W[/tex] as ab + cd is a rational number being the sum and product of rational numbers.

Let [tex]x=ab\varepsilon W[/tex] and k be a scalar

[tex]kx=kab\,\,\varepsilon \,\,W[/tex] being the product of rational numbers a and b.

Therefore, W is a subspace as it satisfies all the conditions.