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Linear Transformations
The main objects of study in any course in linear algebra are linear functions:
Definition A function L : V! W is linear if V and W are vector spaces
and
L(ru + sv) = rL(u) + sL(v)
for all u, v 2 V and r, s 2 R.
Reading homework: problem 1
Remark We will often refer to linear functions by names like “linear map”, “linear operator” or “linear transformation”. In some contexts you will also see the name “homomorphism” which generally is applied to functions from one kind of set to the same kind of set while respecting any structures on the sets; linear maps are from vector spaces to vector spaces that respect scalar multiplication and addition, the two structures on vector spaces. It is common to denote a linear function by capital L as a reminder of its linearity, but sometimes we will use just f , after all we are just studying very special functions.
The definition above coincides with the two part description in Chapter 1;
the case r = 1, s = 1 describes additivity, while s = 0 describes homogeneity.
We are now ready to learn the powerful consequences of linearity.
112 Linear Transformations
6.1 The Consequence of Linearity
Now that we have a su ciently general notion of vector space it is time to
talk about why linear operators are so special. Think about what is required
to fully specify a real function of one variable. One output must be specified
for each input. That is an infinite amount of information.
By contrast, even though a linear function can have infinitely many ele-
ments in its domain, it is specified by a very small amount of information.
Example 69 (One output specifies infinitely many) If you know that the function L is linear and that
L
then you do not need any more information to figure out
L
, L
, L
, L
, etc... ,
because by homogeneity
L
= L
= 5L
In this way an an infinite number of outputs is specified by just one.
Example 70 (Two outputs in R 2 specifies all outputs) Likewise, if you know that L is linear and that
L
and L
then you don’t need any more information to compute
L
because by additivity
L
= L
= L
+ L
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6.2 Linear Functions on Hyperplanes
It is not always so easy to write a linear operator as a matrix. Generally,
this will amount to solving a linear systems problem. Examining a linear
function whose domain is a hyperplane is instructive.
Example 71 Let
V =
c (^1)
A (^) + c (^2)
A
c 1 , c 2 2 R
and consider L : V! R 3 be a linear function that obeys
L
A =
A , L
A =
A .
By linearity this specifies the action of L on any vector from V as
L
(^4) c (^1)
A (^) + c (^2)
A
(^5) = (c 1 + c 2 )
A .
The domain of L is a plane and its range is the line through the origin in the x (^2) direction. It is not clear how to formulate L as a matrix; since
L
c (^1) c 1 + c (^2) c (^2)
A =
A
c (^1) c 1 + c (^2) c (^2)
A (^) = (c 1 + c 2 )
A ,
or
L
c (^1) c 1 + c (^2) c (^2)
A =
A
c (^1) c 1 + c (^2) c (^2)
A (^) = (c 1 + c 2 )
A ,
you might suspect that L is equivalent to one of these 3 ⇥ 3 matrices. It is not. By the natural domain convention, all 3 ⇥ 3 matrices have R 3 as their domain, and the domain of L is smaller than that. When we do realize this L as a matrix it will be as a 3 ⇥ 2 matrix. We can tell because the domain of L is 2 dimensional and the codomain is 3 dimensional. (You probably already know that the plane has dimension 2, and a
6.3 Linear Di↵erential Operators 115
line is 1 dimensional, but the careful definition of “dimension” takes some work; this is tackled in Chapter 11.) This leads us to write
L
(^4) c (^1)
A (^) + c (^2)
A
(^5) = c (^1)
A (^) + c (^2)
A =
A
c (^1) c (^2)
This makes sense, but requires a warning: The matrix
A (^) specifies L so long
as you also provide the information that you are labeling points in the plane V by the two numbers (c 1 , c 2 ).
6.3 Linear Di↵erential Operators
Your calculus class became much easier when you stopped using the limit
definition of the derivative, learned the power rule, and started using linearity
of the derivative operator.
Example 72 Let V be the vector space of polynomials of degree 2 or less with standard addition and scalar multiplication;
V := {a 0 · 1 + a 1 x + a 2 x 2 | a 0 , a 1 , a 2 2 R}
Let (^) dxd : V! V be the derivative operator. The following three equations, along with linearity of the derivative operator, allow one to take the derivative of any 2nd degree polynomial: d dx
d dx x = 1,
d dx x 2 = 2x.
In particular
d dx (a 0 1 + a 1 x + a 2 x 2 ) = a (^0)
d dx 1 + a (^1)
d dx x + a (^2)
d dx x 2 = 0 + a 1 + 2a 2 x.
Thus, the derivative acting any of the infinitely many second order polynomials is determined by its action for just three inputs.
6.4 Bases (Take 1)
The central idea of linear algebra is to exploit the hidden simplicity of linear
functions. It ends up there is a lot of freedom in how to do this. That
freedom is what makes linear algebra powerful.
6.4 Bases (Take 1) 117
sum of multiples and then applying linearity;
L
x y
= L
x + y 2
x y 2
x + y 2
L
x y 2
L
x + y 2
x y 2
x + y 2(x + y)
3(x y) 4(x y)
4 x 2 y 6 x y
Thus L is completely specified by its value at just two inputs.
It should not surprise you to learn there are infinitely many pairs of
vectors from R 2 with the property that any vector can be expressed as a
linear combination of them; any pair that when used as columns of a matrix
gives an invertible matrix works. Such a pair is called a basis for R 2.
Similarly, there are infinitely many triples of vectors with the property
that any vector from R 3 can be expressed as a linear combination of them:
these are the triples that used as columns of a matrix give an invertible
matrix. Such a triple is called a basis for R 3.
In a similar spirit, there are infinitely many pairs of vectors with the
property that every vector in
V =
c 1
A + c 2
A
c 1 , c 2 2 R
can be expressed as a linear combination of them. Some examples are
V =
c 1
A + c 2
A
c 1 , c 2 2 R
c 1
A + c 2
A
c 1 , c 2 2 R
Such a pair is a called a basis for V.
You probably have some intuitive notion of what dimension means (the
careful mathematical definition is given in chapter 11). Roughly speaking,
118 Linear Transformations
dimension is the number of independent directions available. To figure out
the dimension of a vector space, I stand at the origin, and pick a direction.
If there are any vectors in my vector space that aren’t in that direction, then
I choose another direction that isn’t in the line determined by the direction I
chose. If there are any vectors in my vector space not in the plane determined
by the first two directions, then I choose one of them as my next direction. In
other words, I choose a collection of independent vectors in the vector space
(independent vectors are defined in Chapter 10). A minimal set of indepen-
dent vectors is called a basis (see Chapter 11 for the precise definition). The
number of vectors in my basis is the dimension of the vector space. Every
vector space has many bases, but all bases for a particular vector space have
the same number of vectors. Thus dimension is a well-defined concept.
The fact that every vector space (over R) has infinitely many bases is
actually very useful. Often a good choice of basis can reduce the time required
to run a calculation in dramatic ways!
In summary:
A basis is a set of vectors in terms of which it is possible to
uniquely express any other vector.
6.5 Review Problems
Webwork:
Reading problems 1 , 2
Linear? 3
Matrix ⇥ vector 4, 5
Linearity 6, 7
1. Show that the pair of conditions:
L(u + v) = L(u) + L(v)
L(cv) = cL(v)
(valid for all vectors u, v and any scalar c) is equivalent to the single
condition:
L(ru + sv) = rL(u) + sL(v) , (2)
(for all vectors u, v and any scalars r and s). Your answer should have
two parts. Show that (1) ) (2), and then show that (2) ) (1).
120 Linear Transformations
