16.5. EXERCISES 399

is an invertible matrix.

21. Let the field of scalars be Q, the rational numbers and let the vectors be of the forma+ b

√2 where a,b are rational numbers. Show that this collection of vectors is a

vector space with field of scalars Q and give a basis for this vector space.

22. Suppose V is a finite dimensional vector space. Based on the exchange theoremabove, it was shown that any two bases have the same number of vectors in them.Give a different proof of this fact using the earlier material in the book. Hint: Sup-pose {x1, · · · ,xn} and {y1, · · · ,ym} are two bases with m< n. Then define φ :Fn 7→Vand ψ : Fm 7→ V by φ (a) ≡ ∑

nk=1 akxk, and ψ (b) ≡ ∑

mj=1 b jy j. Consider the linear

transformation, ψ−1 ◦φ . Argue it is a one to one and onto mapping from Fn to Fm.Now consider a matrix of this linear transformation and its row reduced echelonform.

23. This and the following problems will present most of a differential equations course.To begin with, consider the scalar initial value problem

y′ = ay, y(t0) = y0

When a is real, show the unique solution to this problem is y = y0ea(t−t0). Nextsuppose

y′ = (a+ ib)y, y(t0) = y0 (16.8)

where y(t) = u(t)+ iv(t) . Show there exists a unique solution and it is

y(t) = y0ea(t−t0) (cosb(t− t0)+ isinb(t− t0))≡ e(a+ib)(t−t0)y0. (16.9)

Next show that for a real or complex there exists a unique solution to the initial valueproblem

y′ = ay+ f , y(t0) = y0

and it is given by

y(t) = ea(t−t0)y0 + eat∫ t

t0e−as f (s)ds.

Hint: For the first part write as y′− ay = 0 and multiply both sides by e−at . Thenexplain why you get

ddt

(e−aty(t)

)= 0, y(t0) = 0.

Now you finish the argument. To show uniqueness in the second part, suppose

y′ = (a+ ib)y, y(0) = 0

and verify this requires y(t) = 0. To do this, note

y′ = (a− ib)y, y(0) = 0

and that

ddt|y(t)|2 = y′ (t)y(t)+ y′ (t)y(t) = (a+ ib)y(t)y(t)+(a− ib)y(t)y(t)

= 2a |y(t)|2 , |y|2 (t0) = 0