Determine whether or not the given set is a subspaceSolution

Determine whether or not the given set is a subspace

Solution

1. Set x=0 ie 0 vector

A*0+2*0=0=B*0

Hence, 0 belongs to W

2.

Let, x ,y belong to W

Hence, Ax+2x=Bx

Ay+2y=By

Ax+2x+Ay+2y=Ax+Ay+2x+2y=A(x+y)+2(x+y)=Bx+By=B(x+y)

Hence, (x+y) belongs to W

3.

Let c be a real number and x be in W

Ax+2x=Bx

Hence, c(Ax+2x)=c(Bx)

cAx+2cx=B(cx)

A(cx)+2(cx)=B(cx)

Hence, cx belongs to W

Hence W is a subspace of Rn