Determine if the following is a subspace Is the given set a

Determine if the following is a subspace

Is the given set a subspace of R³? Justify your answer. H={[2a, b, 3c]: for a,b, and c rational numbers}

Solution

Yes it is a subspace.

1. 0 belongs to this subspace.

2. Let, X and Y be in this subspace so that:

X=[2a,b,3c]

Y=[2p,q,3r]

X+Y=[2(a+p),b+q,3(c+r)]

Hence, X+Y is also in H

3.

X=[2a,b,3c] be in H and c =sqrt{2}

Y=sqrt{2}X=[2sqrt{2}a,sqrt{2}b,sqrt{2}c] which is not in H as entries of Y are irrational.