In Exercises 1−8,W is a subset of R
2
consisting of vec- 7. W={x:x
1
2
+x
2
=1} tors of the form 8. W={x:x
1
x
2
=0} x=[
x
1
x
2
].
In Exercises 9-17, W is a subset of R
3
consisting of
vectors of the form
In each case determine whether W is a subspace of R
2
. If W is a subspace, then give a geometric description of W. 1. W={x:x
1
=2x
2
} 2. W={x:x
1
−x
2
=2} 3. W={x:x
1
=x
2
or x
1
=−x
2
} 4. W={x:x
1
and x
2
are rational numbers } In each case, determine whether W is a subspace of R
3
. If W is a subspace, then give a geometric description 5. W={x:x
1
and of W. 6. W={x:∣x
1
∣+∣x
2
∣=0} 9. W={x:x
3
=2x
1
−x
2
} 10. W={x:x
2
=x
3
+x
1
} 11. W={x:x
1
,x
2
=x
3
} 3.2 Vector Space Properties of R
n
175 12. W={x:x
1
=2x
3
} 26. lnR
4
, let x=[1,−3,2,1]
T
,y=[2,1,3,2]
T
, and 13. W={x:x
1
2
=x
1
+x
2
} z=[−3,2,−1,4]
T
. Set a=2 and b=−3. Illus- trate that the ten properties of Theorem 1 are satisfied 14. W={x:x
2
=0} by x,y,z,a, and b. 15. W={x:x
1
=2x
3
,x
2
=−x
3
} 27. In R
2
, suppose that scalar multiplication were de- 16. W={x:x
3
=x
2
=2x
1
} fined by 17. W={x:x
2
=x
3
=0} 18. Let a be a fixed vector in R
3
, and define W to be the subset of R
3
given by ax=a[
x
1
x
2
]=[
2ax
1
2ax
2
] W={x:a
T
x=0}. for every scalar a. Illustrate with specific examples Prove that W is a subspace of R
3
. those properties of Theorem 1 that are not satisfied.