contestada

Is W is a subspace of V? If not, state why. Assume that V has the standard operations. (Select all that apply.)
W = {(x1, x2, x3, 0): x1, x2, and x3 are real numbers}
V = R4

A) W is a subspace of V.

B) W is not a subspace of V because it is not closed under addition.

C) W is not a subspace of V because it is not closed under scalar multiplication.

Respuesta :

fortuneonyemuwa

Answer:

A) W is a sub space of V

Step-by-step explanation:

given that [tex]V=R^4[/tex] and [tex]W= [(x_1,x_2,x_3,0)|x_1,x_2,x_3E..R] [/tex]

W is a subspace of V because of the follwing.

let [tex](x_1,x_2,x_3,0),(y_1,y_2,y_3,0)[/tex] ∈ [tex]W[/tex] and [tex]a,b[/tex] ∈ [tex]V [/tex]

now [tex]a(x_1,x_2,x_3,0)+b(y_1,y_2,y_3,0)= (ax_1, ax_2,ax_3,0)+(by_1, by_2,by_3, 0)\\\\=(ax_1, +by_1,ax_2+by_2,ax_3,+by_3,0+0)=(ax_1+by_1, ax_2+by_2,ax_3, +by_3,0)[/tex]∈ [tex]W[/tex]

since [tex]ax_1 +by_1,ax_2 ,+by_2,ax_3+by_3[/tex] are all elements of R

W is closed under vector adiition and scalar multiplication.

Hence W is a sub space of V.