16.5. EXERCISES 397

By the above theorem, this could not happen if π were algebraic because then iπ wouldalso be algebraic. Recall these algebraic numbers form a field and i is clearly algebraic,being a root of x2 + 1. This fact about π was first proved by Lindemannn in 1882 andthen the general theorem above was proved by Weierstrass in 1885. This fact that π istranscendental solved an old problem called squaring the circle which was to construct asquare with the same area as a circle using a straight edge and compass. It can be shownthat the fact π is transcendental implies this problem is impossible.1

16.5 Exercises1. Let M =

{u = (u1,u2,u3,u4) ∈ R4 : |u1| ≤ 4

}. Is M a subspace? Explain.

2. Let M ={

u = (u1,u2,u3,u4) ∈ R4 : sin(u1) = 1}. Is M a subspace? Explain.

3. If you have 5 vectors in F5 and the vectors are linearly independent, can it always beconcluded they span F5? Here F is an arbitrary field. Explain.

4. If you have 6 vectors in F5, is it possible they are linearly independent? Here F is anarbitrary field. Explain.

5. Show in any vector space, 0 is unique. This is done in the book. You do it yourself.

6. ↑In any vector space, show that if x+y = 0, then y =−x.This is done in the book.You do it yourself.

7. ↑Show that in any vector space, 0x = 0. That is, the scalar 0 times the vector x givesthe vector 0. This is done in the book. You do it yourself.

8. ↑Show that in any vector space, (−1)x =−x.This is done in the book. You do ityourself.

9. Let X be a vector space and suppose {x1, · · · ,xk} is a set of vectors from X . Showthat 0 is in span(x1, · · · ,xk) .This is done in the book. You do it yourself.

10. Let X consist of the real valued functions which are defined on an interval [a,b] . Forf ,g ∈ X , f +g is the name of the function which satisfies ( f +g)(x) = f (x)+g(x)and for α a real number, (α f )(x)≡ α ( f (x)). Show this is a vector space with fieldof scalars equal to R. Also explain why it cannot possibly be finite dimensional.

11. Let S be a nonempty set and let V denote the set of all functions which are defined onS and have values in W a vector space having field of scalars F. Also define vectoraddition according to the usual rule, ( f +g)(s) ≡ f (s)+ g(s) and scalar multipli-cation by (α f )(s) ≡ α f (s). Show that V is a vector space with field of scalarsF.

12. Verify that any field F is a vector space with field of scalars F. However, show thatR is a vector space with field of scalars Q.

1Gilbert, the librettist of the Savoy operas, may have heard about this great achievement. In Princess Ida whichopened in 1884 he has the following lines. “As for fashion they forswear it, so the say - so they say; and the circle- they will square it some fine day some fine day.” Of course it had been proved impossible to do this a couple ofyears before.