Question

Let V be a vector space over a filed F and x,y,z is an element of V then show that the set of all liner combinations of x,y and z,W=(ax+by+cz:a,b,c are an element of F) is a subspace of V over F. This subspace is called the span of (x,y,z)

Answer 1

A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V .

then:

$W_1+W_2=a(x_1+x_2)+b(y_1+y_1)+c(z_1+z_2)$ is in W

$kW=k(ax+by+cz)=kax+kby+kcz$ is in W

so, W is a subspace of V over F