For each of the following subsets of F3 determine whether it

For each of the following subsets of F3, determine whether it is a sub- space of F3 1

Solution

c)

False

Consider two vectors: (0,1,1),(1,1,0)

0*1*1=0,1*1*0=0

Hence both vectors are in the set

Sum of the two vectors is

(0,1,1)+(1,1,0)=(1,2,1) which is not in the set

Hence it is not a subspace and in subspace sum of any two elements must also be in the set

d)

1. 0 vector belongs to this set because:0=5*0

2. Let, (x1,x2,x3),(y1,y2,y3) belong to this set

So,

x1=5x3,y1=5y3

Sum of vectors is

(x1,x2,x3)+(y1,y2,y3)=(x1+y1,x2+y2,x3+y3)

x1+y1=5x3+5y3=5(x3+y3)

Hence sum is also in the set

3. Let, (x1,x2,x3) be in the set and c be real

So, x1=5x3

cx1=c(5x3)=5(cx3)

Hence ,c(x1,x2,x3) is in the set

Hence it is a subspace