18.2 Matrices and Linear Transformations
Consider the following system of equations.
From now on, we will write vectors as column vectors and if we want to consider a row
vector, we write it as xT. Thus you would have
Then the above system can be written as
We write the above as follows:
Obviously, we could do something similar for any system of linear equations. Thus, a
simpler way to write Definition 18.1.19 is as follows. A linear system of equations having
m equations and n variables is of the form
where A is the matrix having the columns ai given above. It is an m×n matrix because
it has m rows and n columns. The right side b is m× 1. The idea is that for x a vector,
Ax is another vector and you want to choose x in such a way that Ax = b. The matrix A
has an entry Aij which is in the ith row and jth column. Then another way to write 18.9
or more simply,
A matrix is a rectangular array of numbers as in 18.9. In this book, they will
be real numbers. However, nothing is changed if they are complex numbers.
To emphasize this fact, I will sometimes write F to indicate the name of the
numbers used. Thus F could be either ℝ or the complex numbers ℂ. Several of
these rectangular arrays are referred to as matrices. For example, here is a
The size or dimension of a matrix is defined as m×n where m is the number of rows and
n is the number of columns. The above matrix is a 3 × 4 matrix because there are three
rows and four columns. The first row is
the second row is
forth. The first column is
. When specifying the size of a matrix,
you always list the number of rows before the number of columns. Also, you
can remember the columns are like columns in a Greek temple. They stand
upright while the rows just lie there like rows made by a tractor in a plowed
field. Elements of the matrix are identified according to position in the matrix.
For example, 8 is in position 2,
3 because it is in the second row and the third
column. You might remember that you always list the rows before the columns
by using the phrase Row
atholic. The symbol,
refers to a matrix.
The entry in the
row and the jth
column of this matrix is denoted by aij
. Using this notation on the above matrix, a23
You think of a matrix as an explicit way to compute a linear transformation. Let
be two n×
1 matrices, vectors in Fn
. Then if a,b
numbers (called scalars in this context),
is going to be an m × 1 vector. Its ith component is by definition
This is what it means to say that matrix multiplication is an example of a linear
transformation. Here is the formal definition of a linear transformation.
Definition 18.2.1 A function T : Fn → Fm is called a linear
transformation if for any scalars a,b and vectors x,y in Fn,T
The important observation about this is the following theorem. First is some standard
Definition 18.2.2 Define ei as the column vector with a 1 in the ith position
from the top and a zero everywhere else. The size of ei will be determined by context.
Now with the above definition, here is an important observation.
Observation 18.2.3 Let x =
T. Then x
Theorem 18.2.4 If T : Fn → Fm is a linear transformation, then there
exists a unique m×n matrix A such that Tx = Ax where on the right the meaning
is matrix multiplication of a vector.
Proof: This follows from the following computation resulting from the fact that T is
Thus there is a matrix A and it must equal
Because of this theorem, we will regard matrices as linear transformations whenever
convenient. This is as far as I will take this because this is not a linear algebra book, but
in linear algebra, one considers the notion of a matrix of a linear transformation in terms
of an arbitrary basis. Bases will be discussed a little later. The big problem in
linear algebra is to determine whether two matrices represent the same linear
The next section gives an example of the usefulness of this concept of representing a
linear transformation in terms of a matrix.