7.4. EXERCISES 209

3. Let M ={u = (u1, u2, u3, u4) ∈ R4 : u3 ≥ u1

}. Is M a subspace? Explain.

4. Let w ∈ R4 and let M ={u = (u1, u2, u3, u4) ∈ R4 : w · u = 0

}. Is M a subspace?

Explain.

5. Let M ={u = (u1, u2, u3, u4) ∈ R4 : ui ≥ 0 for each i = 1, 2, 3, 4

}. Is M a subspace?

Explain.

6. Let w,w1 be given vectors in R4 and define

M ={u = (u1, u2, u3, u4) ∈ R4 : w · u = 0 and w1 · u = 0

}.

Is M a subspace? Explain.

7. Let M ={u = (u1, u2, u3, u4) ∈ R4 : |u1| ≤ 4

}. Is M a subspace? Explain.

8. Let M ={u = (u1, u2, u3, u4) ∈ R4 : sin (u1) = 1

}. Is M a subspace? Explain.

9. Suppose {x1, · · · ,xk} is a set of vectors from Fn. Show that 0 is in span (x1, · · · ,xk) .

10. Consider the vectors of the form 2t+ 3s

s− t

t+ s

 : s, t ∈ R

 .

Is this set of vectors a subspace of R3? If so, explain why, give a basis for the subspaceand find its dimension.

11. Consider the vectors of the form

2t+ 3s+ u

s− t

t+ s

u

 : s, t, u ∈ R

 .

Is this set of vectors a subspace of R4? If so, explain why, give a basis for the subspaceand find its dimension.

12. Consider the vectors of the form

2t+ u+ 1

t+ 3u

t+ s+ v

u

 : s, t, u, v ∈ R

 .

Is this set of vectors a subspace of R4? If so, explain why, give a basis for the subspaceand find its dimension.

13. Let V denote the set of functions defined on [0, 1]. Vector addition is defined as(f + g) (x) ≡ f (x) + g (x) and scalar multiplication is defined as (αf) (x) ≡ α (f (x)).Verify V is a vector space. What is its dimension, finite or infinite? Justify youranswer.