version of 1 February 1997

I.  Linearity.†

This course is concerned with some of the most important methods of applied mathematics, without which many technological developments of the nineteenth and twentieth centuries would have been impossible.  When you have finished this course you will be able to solve most of the partial differential equations and integral equations that you are likely to encounter in engineering, and, just as importantly, you will understand what to expect from the solutions, which might emerge in a mysterious fashion from a computer at your workplace.  You will also learn techniques that are useful in other contexts - Fourier series, special functions, and iteration.

The leitmotifs are linearity and orthogonality.  By this stage in your education, you are familiar with vectors.  You have added them by the head-to-tail construction, or by adding their components.  You also know how to take their scalar (dot) product, producing a number equal to the product of the two lengths and the cosine of the angle between them:

One of the most powerful and astonishing ideas in modern mathematics is that in many regards functions can be treated just like vectors.   There is a very close analogy between sets of functions and spaces of vectors, which will guide us to striking solutions for several of the most important differential equations you will use in science and engineering, especially the wave equation, the heat equation, and the potential equation.

Mathematicians have a way of making analogies without the mushiness that so often afflicts analogies in so many other contexts.  It is called abstraction.  While abstraction may sometimes seem like theory divorced from reality, when used properly it is a wonderful, and very practical, tool for solving problems.

This is why we will begin by making vectors abstract.  Let us recall some facts about everyday vectors.  If we have a bag full of vectors, we can scale them and add them together.  We call the result a linear combination:

	a1v1 +  a2v2 +  a3v3 +  a4v4

We'll normally use Greek letters for scalars (= ordinary real or complex numbers).  It doesn't matter how many are in the combination, but unless we explicitly state otherwise, we will assume that it is only a finite sum, a finite linear combination.  Of course, we can make linear combinations of functions, too:

	a1f1(x) +  a2f2(x) +  a3f3(x) +  ... + anfn(x)

and the result is another function.  In this way, the set of all functions is a vector space.  

Definition I.1.  More formally, a vector space over the complex numbers  is a set of entities, abstractly called vectors, for which 

1.  Any finite linear combination of vectors is a member of the same set
2.  The usual commutative rule holds for addition:  v + w = w + v, 
3.  Just for consistency, the usual commutative, associative, and distributive laws hold for vector addition and scalar multiplication.  In other words
	 a v + a w = a (v+w), 
	a v + bv = (a+b)v, 
	(a b) v = a(b v) .  
	Also, 1 v = v.  

In practice the rules in 3 are obvious and not very interesting.  From these rules, you can show some other properties, such as:

There is a special element, which will be called the zero vector, equal to the scalar 0 times any vector whatsoever.  It has the property that for any vector v, 0 + v = v.

For any vector v, there is a negative vector, called -v, such that v + (-v) = 0 (the zero vector).  

Certain pretty reasonable conventions will be made, such as writing v - w instead of v + (‑w).  Great - all the usual stuff works, right?  Well, not quite.  We don't  assume abstractly that we can multiply vectors by one another.  Things like the dot product and the cross product work for some vector spaces but not for others.  The other deep issue lurking in the shadows is infinite linear combinations, i.e., infinite series.  Fourier liked them, and we'll see lots of them later.  But if the vectors are functions, perhaps the infinite series converges for some x and not for others.  For instance, 
	
when x is a multiple of p, but it is certainly an improper sum when x = p/2, and it is not immediately clear what happens for other values of x.  What does the series mean then?  (We shall address this issue in Chapter III).

Examples I.2.

1.  The usual two-dimensional or three-dimensional vectors.  It makes little difference whether they are thought of as column vectors such as 
	
or as row vectors (2,3).  The set of all such vectors will be denoted C2 or C3 (assuming complex entries are allowed - otherwise R2 or respectively R3).  Mathematica and Maple can perform all of the usual vector operations in a straightforward way.

2.  The set of complex numbers.  Here there is no difference between a vector and a scalar, and you can check all the properties pretty easily.  We call this a one-dimensional vector space.

3.  The set Cn of n numbers in a list.  These are manipulated just like 2- or 3- vectors, except that the number of components is some other fixed number, n.  For instance, with C4,  we might have elements such as (1,2,3,4) and (1,0,-1,-2), which can be added and multiplied as follows: 

    (1,2,3,4) + (1,0,-1,-2) = (2, 2, 2, 2) 
    (1,2,3,4) . (1,0,-1,-2) = -10, 
etc. 
4.  The set of continuous functions of a variable x, 0 < x < 1.  The rather stupid function f0(x) = 0 for all x plays the rôle of the zero element.

5.  A smaller vector space of functions.  Instead of simply listing n numbers, let us multiply them to define another vector space of functions.  For example, for some fixed n, consider 
	a1 sin(x) + a2 sin(2x) + ... + an sin(nx),
where the numbers ak can take on any value.  Notice that this vector space is a part of the one of example 4.  In other words, it is a subspace .

6.  The set of, say, 2¥3 matrices.  Addition means addition component-wise:

and scalar multiplication affects all components:

7.  The set of 3-component vectors (x,y,z) such that x - 2y + z = 0.  This is a plane through the origin.

Definitions I.3.  A set of vectors {v1, ..., vn} is linearly independent  if it is impossible to write any of the vectors in the set as a linear combination of the rest.  Linearly dependent is the opposite notion.  The dimension  of a vector space V is the largest number n of linearly independent vectors {v1, ..., vn} which are contained in V.  The span  of a set {v1, ..., vn} is the vector space V obtained by considering all possible linear combinations from the set.  We also say that {v1, ..., vn} spans  V.  A set {v1, ..., vn} is a basis  for a finite-dimensional vector space V if {v1, ..., vn} is linearly independent and spans V.

Notice that the only way that two vectors can be linearly dependent is for them to be proportional (parallel) or for one of them to be 0.   If the set has the same number of vectors as each vector has components, which frequently is the case, then there is a calculation to test for linear dependence.  Array the vectors in a square matrix and calculate its determinant.  If the determinant is 0, they are dependent, and otherwise they are independent.  For example, consider the vectors {(1,2,3), (4,5,6),(7,8,9)}, which are not obviously linearly dependent.  A calculation shows that

Indeed, we can solve for one of these vectors as a linear combination of the others:

	(7,8,9) =  (-1)  (1,2,3) + 2 (4,5,6)

Many of our vector spaces will be infinite-dimensional.  For example, {sin(n x)} for n = 1, 2, ..., is an infinite, linearly independent set of continuous functions - there is no way to write sin(400 x), for example, as a linear combination of sine functions with lower frequencies, so you can see that each time we introduce a sine function of a higher frequency into the list, it is independent of the sines that were already there.  An infinite linearly independent set is a basis for V is every element of V is a limit of finite linear combinations of the set - but we shall have to say more later about such limits.  A vector space has lots of different bases, but all bases contain the same number of items.

For practical purposes, the dimension of a set is the number of degrees of freedom, i.e., the number of parameters it takes to describe the set.  For example, Cn has dimension n, and the set of 2¥3 matrices has six elements, so its dimension is 6. 

Model problem I.4.  Show that the plane x + y + z = 0 is a two-dimensional vector space.  

The verification of the vector space properties will be left to the reader.  Here is how to show that the dimension is 2:

Solution.  The solution can't be 3, since there are vectors in R3  which are not in the plane, such as (1,1,1).  On the other hand, here are two independent vectors in the plane:  (1,-1,0) and (1,1,-2).  
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
Further observations.  The general vector in the plane can be written with 2 parameters multiplying these two vectors, and it is possible to directly find the formula expressing the general vector this way.  First write the general vector in the plane as (x,y,-x-y) (by substituting for z - notice 2 parameters for 2 dimensions).  It is a straightforward exercise in linear algebra - with or without software - to solve for a and b such that 
	(x,y,-x-y) = a (1,-1,0) + b (1,1,-2).
We can solve this equation with the choices a = (x-y)/2 and b = (x+y)/2.
Hence the two vectors we found in the solution are a basis for the plane.  Any two independent vectors in the plane form a basis.

Definition I.5.  A linear transformation  is a function on vectors, where it doesn't matter whether linear combinations are made before or after the transformation.  In other words, 

      F(a1 v1 +  a2 v2 +  a3 v3 +  a4 v4) 
                 = a1F(v1) +  a2F(v2) +  a3F(v3) +  a4F(v4)

Linear transformations are also called linear operators, or just operators for short.  You know plenty of examples:

Examples I.6.
1.  Matrices.  If M is a matrix and v, w etc. are column vectors, then 

	M(a1v1 +  a2v2 +  a3v3 +  a4v4) 
                = a1Mv1 +  a2Mv2 +  a3Mv3 +  a4Mv4

Think of rotation and reflection matrices here.  If you put a bunch of vectors head-to-tail and rotate the assemblage, or look at it in a mirror, you get the same effect as if you first rotate or reflect the vectors, and then put them  head-to-tail.  It may be less geometrically obvious when the matrix distorts vectors in a trickier way, or there are more than three dimensions, but it is still true.   In this example, algebraic intuition might be more convincing than geometric intuition
2.  Derivatives and integrals.  As we know, 

	 

3.  This may seem silly at first, but the identity operator  Id, which just leaves functions alone is a linear transformation:
	Id(a1 f1 +  a2 f2) = a1 Id f1 +  a2 Id f2, 

since both sides are just round-about ways of writing a1 f1 +  a2 f2.  The identity operator is a useful bit of notation for much the same reason as the identity matrix, 

the effect of which on any vector is to leave it unchanged.

A linear transformation is a function defined on vectors, and the output is always a vector, but the output need not be the same kind of vector as the input.  You should be familiar with this from matrices, since a 2¥3 matrix acts on 3-vectors and produces 2-vectors.  Similarly, the operator D acts on a vector space of functions assumed to be differentiable and produces functions which are not necessarily differentiable.  

Example I.7.  More exotic possibilities are possible, such as the operator which acts on 2¥2 matrices by the rule:

The whole theory we are going to develop begins with the analogy between Example I.3.1 and Example I.3.2.  We can think of the linear operator  of differentiation by x as a kind of abstract matrix, denoted D .  If we also think of the functions f and g as entities unto themselves, without focusing on the variable x, then the expression in the first part of Example I.3.2 looks very much like Example I.3.1:

	D(a1f1 +  a2f2) = a1Df1 +  a2Df2.

The custom with linear transformations, as with matrices, is to do without the parentheses when the input variable is clear.  It is tempting to manipulate linear operators in many additional ways as if they were matrices, for instance to multiply them together.  This often works.  For instance D2 = D D can be thought of as the second derivative operator, and expressions such as D2 + D + 2 Id make sense.  In passing, notice that if S is the integration operator of the second part of Example 2, then D S f = f, so D is an inverse to S in the sense that D S = Id.  (But is S D = Id?)

Linear ODE's.  In your course on ordinary differential equations, you studied linear differential equations, a good example of which would be

	D2y + Dy + 2y = 0

More specifically, this is an example of a "linear, homogeneous differential equation of second order, with constant coefficients".  We can picture this equation as one for the null space  of a differential operator  A:  

	A y := (D2 + D + 2 Id) y = 0.                                               (1.1)

By definition, the null space N(A) is the set of all vectors solving this equation; some texts refer to it as the kernel  of A.  There is no difference.  You may remember that the null space of a matrix is always a linear subspace, and the same is true for the null space of any linear operator.   E. g., the plane of Example 1.2.6 is the null space of the matrix

						(1.2)

What does this abstract statement about subspaces mean in a concrete way for the problem of solving a linear homogeneous problem?  It is the famous superposition principle:  If we can find two (or more) solutions of a homogeneous equation, then any linear combination of them is also a solution.  

For matrices the general solution of the homogeneous equation is the set of linear combinations of a finite number of particular solutions.  This will also be true for linear ordinary differential equations (the situation is a little more complicated for linear partial differential equations).  Indeed, the number of independent functions in the null space of an ordinary differential operator is equal to its order (the highest power of D).  This is true even when the coefficients are allowed to depend on x, so long as there are no singularities (such as values of x where the coefficients become 0 or ∞).

Let us illustrate the situation with the two operators mentioned above.

Model Problem I.8.  Find the null space of M as given in (1.2), which is the same as finding the general solution of M v = 0.  

In a linear algebra class you probably learned several techniques for finding the solution vectors v, and the way you describe the null space may depend on the technique chosen.  We may prefer to use software, since Maple and Mathematica have commands to find the 
null space of a matrix.  

Solution.

The nullspace can be found with software as a formula relating x, y, and z: 

	x = 2 y - z

Equivalently, this is the plane satisfying   x - 2y + z = 0.  The plane passes through the origin, so it must be true that any linear combination of vectors in this plane again lies in the plane.  (Why is it important that the plane passes through the origin?)  A second way to describe the plane is to find a basis, i.e., two independent vectors such as  
	v1 = (1,1,1) and v2 = (1,0,-1)
which are both in the plane and which form a basis for it.

Model Problem I.9.  Find the general solution of
	A u(x) = 0.                                                                    (1.3)

Solution.  The method here is to guess a solution of the form emx, and substitute to find what m must be.  Usually, we find two possible values of m, and in a class on ordinary differential equations we learned that the general solution can be obtained from any two linearly independent solutions (for second order ODE's).  If u(x) is of the form emx, then   A u = (m2 + m + 2) u, so if u is not the zero function, we must have 

Thus, two linearly independent solutions to (1.3) are 

As these are complex valued - remembering Euler's formula that 
	exp(i g) = cos(g) + i sin(g)
- they are not necessarily in the most convenient form.  However, they can be recombined easily into a pair of independent real functions, 

The general solution is the span of these two functions, i.e., 

	a1u1(x) + a2u2(x).

Finally, let us recall the solution of nonhomogeneous linear problems, and observe how similarly it looks for matrix equations and differential equations.

Model Problem I.10.  Find the general solution of

Solution.  The first step is to find a particular solution .A bit of trial and error leads to, perhaps, 
	
There are many other, equally good choices for vp.  The general solution is the set of vectors of the form
	
so we see that the homogeneous solution is just added onto vp.  The matrix M annihilates the terms aj vj, so 
	
The explicit answer is

Compare with the following:
Model Problem I.11.  Find the general solution of
	A u = exp(2x).

Solution.  Again, the first step is to find a particular solution.  Trial and error with the guess up(x) = C exp(2 x) leads to
	A up = (4 + 2 + 2) C exp(2x),
showing us that we must choose C = 1/8, making  
The general solution is 
	
where u1,2 are as given above.  
_____________________________________________________________________
Exercises.

Exercise I.1.  For fluency with abstract vectors, use Definition I.1 to derive the rules that followed the definition.

Exercise I.2.  Verify that the spaces in Examples I.2 are all vector spaces.

Exercise I.3.  Find which of the following are vector spaces and which are not.  In all cases, define addition and scalar multiplication in the standard way.

a)  The plane of vectors (x,y,z) such that x - 2y + z = 1.

b)  The set consisting of the zero function and all functions for 0 < x < 1 which are not continuous at 1/2.

c)  The set of 2 by 2 matrices with determinant 0.  

d)  The set of 2 by 2 matrices with trace 0.  (Recall that the trace of a matrix is the sum of all of its diagonal elements from upper left to lower right.)

e)  The set of 3 by 3 matrices with determinant 0.

f)  The set of all polynomials in the variable x.

Exercise I.4.  Show that a set {v1, ..., vn} is linearly independent if and only if the statement that    a1 v1 +  a2 v2 + ...  an vn = 0, implies that all the  a's are zero.

Exercise I.5.  With the operators D and S as defined in the text, acting on the vector space consisting of all polynomials, is S D = Id?  If not, give a simple description of the operator SD.
 
Exercise I.6.   (For review of ordinary differential equations.)  Find the general solutions of the following ordinary differential equations.
a)	u'(t) = - 4 u(t)
b)	u''(x) + (1/x) u'(x) - 9 u(x)/x2 = 0
c)	u'''(x) = u(x)

Exercise I.7.  Find the null space of the matrix 

and find the general solution of 

Exercise I.8.  Find the null space of the differential operator 

	A = D2 - 2 D + Id
and find the general solution of 
	A u(x) = cos(x).