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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.

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